Calculate the derivative of the following function and simplify.

y = [tex]e^{x} csc x[/tex]

Answer:

If the wholesale price of the king-size mattress is $359.00, and the specialty store marks up the price by 40%, the price Debra would pay at the specialty store is:

Wholesale price + Mark-up amount = Price at specialty store

$359.00 + 40% of $359.00 = $359.00 + $143.60 = $502.60

Therefore, Debra would pay $502.60 for the mattress at the specialty store.

The derivative of f(x) is

[tex]f'(x) = -5pi x^(pi-1) + 6.1 * 5.1x^(5.1-1) + 5.1pi^(5.1-1)[/tex].

The derivative of a function is a measure of how that function changes as its input changes. Derivatives are also used in calculus to find the area under a curve, or to solve differential equations.

In this case, the function f(x) is a polynomial, which means it is a combination of terms of the form [tex]ax^b[/tex], where a and b are constants. The derivative of f(x) can be calculated by taking the derivative of each term in the function and then combining them together.

The derivative of a term [tex]ax^b[/tex] is [tex]abx^(b-1)[/tex]. For the first term of f(x),[tex]-5x^pi[/tex], the derivative is [tex]-5pi x^(pi-1)[/tex]. For the second term, [tex]6.1x^5.1[/tex] the derivative is[tex]6.1 * 5.1x^(5.1-1)[/tex]. For the third term, [tex]pi^5.1[/tex], the derivative is [tex]5.1pi^(5.1-1)[/tex].

Combining these terms together, the derivative of f(x) is

[tex]f'(x) = -5pi x^(pi-1) + 6.1 * 5.1x^(5.1-1) + 5.1pi^(5.1-1)[/tex].

This answer is the derivative of the given function. This is how the function changes as its input changes.

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The derivative of f(x)= [tex]-5x^{\pi}+6.1x^{5.1}+\pi^{5.1}[/tex] is  [tex]-5\pi x^{\pi -1}[/tex]+  [tex]6.1*5.1x^{5.1-1}[/tex] +5.1[tex]\pi^{5.1-1}[/tex] which can be calculated with the power rule.

The derivative of a function is a measure of how that function changes as its input changes. Derivatives are also used in calculus to find the area under a curve, or to solve differential equations.

The derivative of the given function f(x) = [tex]-5x^{\pi}+6.1x^{5.1}+\pi^{5.1}[/tex] can be calculated with the power rule, which states that the derivative of xⁿ is nx⁽ⁿ⁻¹⁾

To calculate the derivative of the given function, we begin by applying the power rule to each term.

The first term is [tex]-5^{\pi }[/tex] which has a derivative of [tex]-5\pi x^{\pi -1}[/tex].

The second term is [tex]6.1x^{5.1}[/tex] which has a derivative of [tex]6.1*5.1x^{5.1-1}[/tex].

The third term is [tex]\pi^{5.1}[/tex], which has a derivative of 5.1[tex]\pi^{5.1-1}[/tex].

Therefore, the derivative of the given function

f(x)= [tex]-5x^{\pi}+6.1x^{5.1}+\pi^{5.1}[/tex] is  [tex]-5\pi x^{\pi -1}[/tex]+  [tex]6.1*5.1x^{5.1-1}[/tex] +5.1[tex]\pi^{5.1-1}[/tex].

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Question:

Compute the derivative of the given function.

f(x) = - [tex]5x^{\pi }[/tex]+[tex]6.1x^{5.1}[/tex]+[tex]\pi^{5.1}[/tex]

​

Answer:

The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:

m = (y2 - y1) / (x2 - x1)

Substituting the values, we get:

m = (5 - (-10)) / (6 - 3) = 15/3 = 5

Now that we have the slope, we can use the point-slope form of a linear equation to write the equation of the line:

y - y1 = m(x - x1)

Substituting the values of m, x1, and y1, we get:

y - (-10) = 5(x - 3)

Simplifying and rearranging the equation, we get:

y + 10 = 5x - 15

y = 5x - 25

Therefore, the equation of the line passing through (3,-10) and (6,5) in slope-intercept form is y = 5x - 25.

Step-by-step explanation:

#trust me bro

Australian sheepdogs' life length follows a uniform distribution between 8 and 14 years.

a) The probability that a sheep dog will live at least 10 years, P( X ≥10) is equals to the 1/3

b) The probability that a sheepdog will live no more than 11 years, P( X ≤11) is equals to the 1/2.

c) The probability that a sheepdog will live between 10 and 13 years, P(10≤X≤13), is equals to the 1/2

In statistics, uniform distribution,is a distribution function where every possible result is equally likely, i.e., the probability of each event occurring is the equal. Here we have the length of Australian sheepdogs life follows a uniform distribution between 8 and 14 years. So, the probability distribution function, pdf for uniform distribution is f(x) = 1/(b - a), a<x<b , a= 8 , b = 14

=> f(x) = 1/(14 - 8) = 1/6

Area of uniform distribution= height × base and base = 14 - 8 = 6

height of uniform distribution is = 1/( Max - Min) = 1/6.

a) the probability that a sheepdog will live at least 10 years, P(X ≥10)= 1 - P(X< 10)

The cumulative distribution function in uniform distribution is P(X ≤ x) = (x − a)/(b − a) so, P( X < 10) = (10 - 8)/(14 - 8)

= 2/6 = 1/3

b) the probability that a sheepdog will live no more than 11 years, P( X ≤11)

= (11 - 8)/( 14 - 8)

= 3/6 = 1/2

c) In uniform distribution, P(c ≤ x ≤ d)

= (d-c)/(b- a)

The probability that a sheepdog will live between 10 and 13 years, P( 10≤X≤13), c = 10 , d = 13 , b = 14 , a = 8

=> [tex]P( 10≤X≤13) = \frac{d-c}{b- a} = \frac{13 - 10}{14 - 8} [/tex]

= 3/6 = 1/2

Hence required probability value is 1/2.

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Complete question:

Uniform Questions:

Australian sheepdogs have a relatively short life. The length of their life follows a uniform distribution between 8 and 14 years. Questions:

a)What is the probability that a sheepdog will live at least 10 years?

b)What is the probability that a sheepdog will live no more than 11 years?

c)What is the probability that a sheepdog will live between 10 and 13 years?

The command that will allow you to set the interface link as preferred over another in OSPF is the "ip ospf cost 10" command. The correct answer is D).

The OSPF protocol uses the concept of cost to determine the best path for routing information to a remote network. The cost of a particular path is calculated based on the bandwidth of the interface and is inversely proportional to it. Therefore, the lower the cost, the more preferred the path is.

To set the preferred link, you can decrease the cost of the desired interface using the "ip ospf cost" command. For example, if you want to set the cost of interface GigabitEthernet0/0 to 10, you can use the following command:

ip ospf cost 10 interface GigabitEthernet0/0

This will make OSPF prefer the specified interface over other interfaces with a higher cost, ensuring that traffic is routed through the preferred link.

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____The given question is incomplete, the complete question is given below:

You need to setup a preferred link that OSPF will use to route information to a remote network. Which command will allow you to set the interface link as preferred over another?

A. ip ospf preferred 10

B. ip ospf priority 10

C. ospf bandwidth 10

D. ip ospf cost 10

To approximate the root of √5 with a precision of 0.0078125, the binary search method must be used. The process is repeated until the desired value is reached.

The Intermediate Value Theorem states that if a continuous function takes on two different values at two different points, then it must take on a value in between those two points.

By using the function f(x) = x²-5, it can be seen that f(2) < 0 and f(3) > 0. This means that there must be a value, 2 < c < 3, such that f(c) = 0.

The next step is to determine if f(2.5) is less than or greater than 0. If it is the same sign as 2.5, the endpoint of the same sign must be used as the new endpoint of the interval.

By using the Intermediate Value Theorem, it was proven that there must be a value between 2 and 3 that satisfies the equation f(x) = x²-5. The midpoint of those two points was used as the first approximation, and by repeating the process of binary search, the desired precision was achieved.

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Answer:

150

Step-by-step explanation:

Using the formulas

A=6a2

V=a3

Solving forA

A=6V⅔=6·125⅔ ≈150

Answer:

Step-by-step explanation:

If the volume of a cube is 125 cm³, it means that each side of the cube measures 5 cm (since 5 x 5 x 5 = 125).

To find the surface area of the cube, we need to calculate the area of each of the six faces and add them together.

The area of each face is simply the length of one side squared (or side x side).

So, the surface area of the cube would be:

6 x (5 cm x 5 cm) = 6 x 25 = 150 cm²

Therefore, the surface area of the cube is 150 cm².

As a sound wave travels, the particles in the medium it travels through vibrate back and forth in the same direction as the wave travels.

As a sound wave travels through a medium, such as air or water, the particles in the medium vibrate back and forth in the same direction as the wave travels.

However, the particles themselves do not actually move along with the wave. Instead, the wave causes a disturbance in the particles, which then passes from one particle to the next, resulting in a wave-like pattern of motion that travels through the medium.

In other words, the sound wave is a disturbance that travels through the medium, causing the particles in the medium to oscillate back and forth around their equilibrium positions.

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To calculate the amount of wire needed to fence Mr. Woodstock's land, you will need to use the formula for the perimeter of a rectangular shape. The formula for perimeter is Length + Width x 2.

Plugging in the given dimensions for Length (36 meters) and Width (16 meters) will give us the following calculation:

36 + 16 x 2 = 88 meters

Therefore, Mr. Woodstock will need 88 meters of wire to fence his land.

The given statement " let v be a vector space. we know that v must contain a zero vector, 0v and the zero vector is unique." is true and proved as an element in vector satisfies the define of zero vector i.e.,

0v + u  = u.

To show that the zero vector is unique, we need to prove that there can be only one element in the vector space that satisfies the definition of a zero vector, namely:

For any vector u in v, 0v + u = u + 0v = u.

To do this, suppose that there exist two distinct zero vectors, 0v and 0'v, such that 0v ≠ 0'v. Then, by the definition of a zero vector, we have:

0v + 0'v = 0'v + 0v = 0'v.

But, by the associative property of vector addition, we can also write:

0v + 0'v = (0v + u) + (-u + 0'v) = u + (-u) = 0v.

Similarly, we can write:

0'v + 0v = (0'v + u) + (-u + 0v) = u + (-u) = 0'v.

These equations show that 0v = 0'v, which contradicts our assumption that 0v ≠ 0'v. Therefore, the zero vector is unique, and there can be only one element in the vector space that satisfies the definition of a zero vector.

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His fortnightly income is $3000 where the yearly salary is $78000 using 26 fortnights in a year.

Fortnightly income is the amount of income a person earns every two weeks. It is usually calculated by dividing the person's yearly income by the number of fortnights in a year, which is typically 26.

To calculate the fortnightly income (F), we need to divide the yearly salary (Y) by the number of fortnights in a year (N), which is 26.

So mathematically, we can express the calculation of the fortnightly income as:

F = Y / N

Substituting the given values, we get:

F = $78000 / 26

Simplifying the expression, we get:

F = $3000

Therefore, his fortnightly income is $3000.

For example, if a person's yearly salary is $52,000, their fortnightly income would be calculated as:

Fortnightly income = Yearly salary / Number of fortnights

Fortnightly income = $52,000 / 26

Fortnightly income = $2,000

The person's fortnightly income would be $2,000.

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The complete question is Tom’s yearly salary is $78000. Calculate Tom’s fortnightly income. (Use 26 fortnights in a year). Fortnightly income = ?$

Answer:

Neither parallel or perpendicular

Step-by-step explanation:

Parallel lines have the same slope.  Perpendicular slopes are the negative reciprocals of each other.

30y +25x = -180  Subtract 25x from both sides

30y + 25x - 25x = -25x - 180

30y = -25x - 180  Divide all the way through by 30

[tex]\frac{30y}{30}[/tex] = [tex]\frac{-25}{30}[/tex] - [tex]\frac{180}{30}[/tex]

y = [tex]\frac{-5}{6}[/tex] - 6

y = [tex]\frac{-6}{5}[/tex] x + 6

Comparing the 2 equations, we have slope of

[tex]\frac{-5}{6}[/tex] and [tex]\frac{-6}{5}[/tex]

These slopes are not the same so they are not parallel.

These slopes are not negative reciprocals. so they are not perpendicular.

Perpendicular slopes would be [tex]\frac{-5}{6}[/tex] and [tex]\frac{6}{5}[/tex] or [tex]\frac{-6}{5}[/tex] and [tex]\frac{5}{6}[/tex].  One slope would have to be negative and the other slope would have to be positive which they are not.

Helping in the name of Jesus.

The point estimate for the population mean age at first marriage is 21.89, the true population mean age at first marriage falls between 21.815 and 21.965 years with a small margin of error due to a large sample size. A 99% confidence interval is (21.836, 21.944).

The point estimate at first marriage is 21.89.

We can interpret the 99% confidence interval as follows: we are 99% confident that the true population mean age at first marriage falls between 21.815 and 21.965 years.

To compute a 95% confidence interval, we can use the formula:

Margin of error = z*(SE)

where z is the z-score corresponding to the desired confidence level (1.96 for 95% confidence), and SE is the standard error of the mean, which is equal to the standard deviation divided by the square root of the sample size.

Thus, for the given data:

Margin of error = 1.96*(4.787/sqrt(26920)) = 0.054

The 95% confidence interval can be computed as:

21.89 ± 0.054

which gives us a range of (21.836, 21.944).

The margin of error for this confidence interval is small because the sample size is very large (n=26920). As the sample size increases, the standard error of the mean decreases, which in turn reduces the margin of error.

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_____The given question is incomplete, the complete qustion is given below:

a. Use the summary to determine the point estimate of the population mean and margin of error for the confidence interval

b. interpret the confidence interval

c. verify the results by computing a 95% confidence interval with the information provided

d. why is the margin of error for this confidence interval so small? A study asked respondents, "If ever married, how old were you when you first married? The results are summarized in the technology excerpt that follows. Complete parts (a) through (d) below. One-Sample T: AGEWED Variable N Mean StDev SE Mean 99.0% CI AGEWED 26920 21.890 4.787 0.029 (21.815, 21.965) L attention and maintarhaan Hansen

In response to the stated question, we may state that The margin of error function is equal to the highest mistake on the estimate.

In mathematics, a function is a connection between two sets of numbers in which each member of the first set (known as the domain) corresponds to a single element in the second set (called the range). In other words, a function takes inputs from one set and produces outputs from another. Inputs are commonly represented by the variable x, whereas outputs are represented by the variable y. A function can be described using an equation or a graph. The equation y = 2x + 1 represents a linear function in which each value of x yields a distinct value of y.v

(a) The population distribution must be assumed to be normal before generating a confidence interval.

(b) The margin of error with 90% confidence is provided by:

Error Margin = Z (/2) * (/n)

Where Z (/2) is the confidence level/2 crucial value, is the population standard deviation (unknown), and n is the sample size.

Error Margin = t (/2, n-1) * (s/n)

Where t (/2, n-1) is the critical value for the degrees of freedom /2 and n-1, and s is the sample standard deviation.

(d) The margin of error is equal to the highest mistake on the estimate.

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Hence the conversion of given numbers according to given demand is

a) N=217 in binary = 11011001

b) N=99 in hexadecimal = 63

c) N= 344 in hexadecimal = 158

d) N= 136 in base 7 = 253.

e) N= 542 in base 5 = 4132.

f) N= 727 in base 8= 1327

g) N= 171 in hexa =1011

h) N=91 in base 3=10101.

i) N= 840 in base 9 =  1133

a) A good method to convert a decimal number to binary is dividing it by 2 and using the remainder of the division as the converted number, starting by the most significant bit (the right one). We can't divide anymore. So we have:

217÷2 = 108 + 1

108÷2 = 54 + 0

54÷2 = 27 + 0

27÷2 = 13 + 1

13÷2 = 6 + 1

6÷2 = 3 + 0

3÷2 = 2 +1

2÷2 = 1

The binary equivalent to 217 is 11011001

b) To convert a number from decimal to hex we can divide the number by 16, taking out the decimal part and multiplying it by 16 using that as our most significant number while using the result of the original division to continue our conversion. So we have:

99÷16 = 6.1875

The decimal part is 0.1875, we multiply it by 16 and obtain 3 as our most significant number. Since we can't divide 6 by 16 we have that as our least significant number then the hexadecimal equivalent is 63.

c) We follow the same steps as in item b:

344÷16 = 21.5

The most significant number is 0.5*16 = 8

21÷16 = 1.3125

The next number is 0.3125*16 = 5

Since we can't divide it anymore we have our result which is 158 in hex.

d) To convert from decimal to base 7 we'll use the same method as to hex, but this time dividing and multiplying by 7.

136÷7 = 19.428571

The most significant number is 0.428571 * 7 = 3

19÷7 = 2.71428571

The next number is 0.71428571*7 = 5

Since we can't divide it anymore we have our result which is 253.

e) To convert from decimal to a base 5 we'll use the same method as before but dividing and multiplying by 5.

542÷5 = 108.4

The most significant number is 0.4*5 = 2

108÷5 = 21.6

The next number is 0.6*5 = 3

21÷5 = 4.2

The next number is 0.2*5 = 1

Since we can't divide it anymore we have our result which is 4132.

f) To convert from decimal to a base 8 we'll use the same method as before but dividing and multiplying by 8.

727÷8 = 90.875

The most significant number is 0.875*8 = 7

90÷8 = 11.25

The next number is 0.25*8 = 2

11÷8 = 1.375

The next number is 0.375*8 = 3

Since we can't divide anymore we have our result which is 1327

g) Following the same steps as before:

171÷16 = 10.6875

The most significant number is 0.6875*16 = 11

Since we can't divide anymore we have our result which is 1011

h) Following the same steps as before:

91÷3 = 30.333333333

The most significant number is 0.333333*3 = 1

30÷3 = 10

The next number is 0

10÷3 = 3.3333333333

The next number is 0.333333*3 = 1

3÷3 = 1

Since we have the final value remainder as 0 the least significant number is 1

Since we can't divide anymore we have our result which is 10101.

i) Following the same steps as before:

840÷9 = 93.333333

The most significant number is 0.33333*9 = 3

93÷9 =  10.3333333

The next number is 0.333333*9 = 3

10÷9 = 1.11111111

The next number is 0.11111111*9 = 1

Since we can't divide anymore we have our result which is 1133

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Answer:

Let's denote the number of adult tickets sold by "x" and the number of children's tickets sold by "y".

We know that the total number of tickets sold is 960, so:

x + y = 960

We also know the total amount collected, which is $11,920:

15x + 11y = 11,920

Now we have two equations with two unknowns, and we can solve for x and y. One way to do this is by using substitution. We can solve the first equation for x:

x = 960 - y

Then substitute this expression for x into the second equation:

15(960 - y) + 11y = 11,920

Expanding the brackets gives:

14,400 - 15y + 11y = 11,920

Simplifying:

4y = 2,480

y = 620

Now we can use this value of y to find x:

x = 960 - y = 960 - 620 = 340

Therefore, 340 adult tickets and 620 children's tickets were sold.

Answer:

Let's use a system of equations to solve the problem.

Let x be the number of adult tickets sold and y be the number of children's tickets sold.

We know that the total number of tickets sold is 960, so we can write:

x + y = 960

We also know that the total revenue from the ticket sales was $11,920. The revenue from the adult tickets is $15 times the number of adult tickets sold, and the revenue from the children's tickets is $11 times the number of children's tickets sold. So we can write:

15x + 11y = 11,920

We now have two equations with two unknowns, which we can solve using substitution or elimination. Let's use elimination:

Multiply the first equation by 11 to get 11x + 11y = 10,560

Subtract the second equation from the first to get:

15x + 11y - 15x - 11y = 11,920 - 10,560

Simplifying, we get:

4x = 1,360

Dividing both sides by 4, we get:

x = 340

So 340 adult tickets were sold.

Substituting this value back into the first equation, we get:

340 + y = 960

Solving for y, we get:

y = 620

So 620 children's tickets were sold.

Therefore, there were 340 adult tickets sold and 620 children's tickets sold.

Answer:

the answer to that is -31

To satisfy the inequality x less-than StartFraction 7 over 12 EndFraction.

Inequalities are called as the mathematical expressions in which both sides are nonequal. Unlike to equations, we compare two values in inequality. Less than (or less than or equal to), greater than (or greater than or equal to), or not equal to signs can be used in place of the equal sign in between.

The inequality is 1/4 + x < 5/6 in order to solve this inequality we need to isolate the value of x, that is our variable of interest. This is shown bellow:

1/4 + x < 5/6

x < 5/6 - 1/4

LMC is used to subtract the fractions we have as follows:

x < (2*5 - 3*1)/12

x < (10 - 3)/12

x< 7/12

The inequality must be satisfied for x to be smaller than 7/12.

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Answer:  x < 7/12

Step-by-step explanation:

Answer:

1.066 (3 d.p.)

Step-by-step explanation:

The volume of the solid formed by revolving a region, R, around a vertical axis, bounded by x = a and x = b, is given by:

[tex]\displaystyle 2\pi \int^b_ar(x)h(x)\;\text{d}x[/tex]

where:

[tex]\hrulefill[/tex]

We want to find the volume of the solid formed by revolving a region, R, around the y-axis, where R is bounded by:

[tex]y=\dfrac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{3}}[/tex]

[tex]y=0[/tex]

[tex]x=0[/tex]

[tex]x=1[/tex]

As the axis of rotation is the y-axis, r(x) = x.

Therefore, in this case:

[tex]r(x)=x[/tex]

[tex]h(x)=\dfrac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{3}}[/tex]

[tex]a=0[/tex]

[tex]b=1[/tex]

Set up the integral:

[tex]\displaystyle 2\pi \int^{1}_0x \cdot\dfrac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{3}}\;\text{d}x[/tex]

Take out the constant:

[tex]\displaystyle 2\pi \cdot \dfrac{1}{\sqrt{2\pi}}\int^{1}_0x \cdot e^{-\frac{x^2}{3}}\;\text{d}x[/tex]

[tex]\displaystyle \sqrt{2\pi}\int^{1}_0x \cdot e^{-\frac{x^2}{3}}\;\text{d}x[/tex]

Integrate using the method of substitution.

[tex]\textsf{Let}\;u=-\dfrac{x^2}{3}\implies \dfrac{\text{d}u}{\text{d}x}=-\dfrac{2x}{3}\implies \text{d}x=-\dfrac{3}{2x}\;\text{d}u[/tex]

[tex]\textsf{When}\;x=0 \implies u=0[/tex]

[tex]\textsf{When}\;x=1 \implies u=-\dfrac{1}{3}[/tex]

Rewrite the original integral in terms of u and du:

[tex]\displaystyle \sqrt{2\pi}\int^{-\frac{1}{3}}_0x \cdot e^{u}\cdot -\dfrac{3}{2x}\;\text{d}u[/tex]

[tex]\displaystyle \sqrt{2\pi}\int^{-\frac{1}{3}}_0 -\dfrac{3}{2}e^{u}\; \text{d}u[/tex]

[tex]-\dfrac{3\sqrt{2\pi}}{2}\displaystyle \int^{-\frac{1}{3}}_0 e^{u}\; \text{d}u[/tex]

Evaluate:

[tex]\begin{aligned}-\dfrac{3\sqrt{2\pi}}{2}\displaystyle \int^{-\frac{1}{3}}_0 e^{u}\; \text{d}u&=-\dfrac{3\sqrt{2\pi}}{2}\left[ \vphantom{\dfrac12}e^u\right]^{-\frac{1}{3}}_0\\\\&=-\dfrac{3\sqrt{2\pi}}{2}\left[ \vphantom{\dfrac12}e^{-\frac{1}{3}}-e^0\right]\\\\&=-\dfrac{3\sqrt{2\pi}}{2}\left[ \vphantom{\dfrac12}e^{-\frac{1}{3}}-1\right]\\\\&=1.06582594...\\\\&=1.066\; \sf (3\;d.p.)\end{aligned}[/tex]

Therefore, the volume of the solid is approximately 1.066 (3 d.p.).

[tex]\hrulefill[/tex]

[tex]\boxed{\begin{minipage}{3 cm}\underline{Integrating $e^x$}\\\\$\displaystyle \int e^x\:\text{d}x=e^x(+\;\text{C})$\end{minipage}}[/tex]

Answer:

a) The middle 95 percent of all miniature Tootsie Rolls will fall between 3.04 grams and 3.56 grams.

b) 6.18% probability that a randomly chosen miniature Tootsie Roll will weigh more than 3.50 grams.

c) 52.29% probability that a randomly chosen miniature Tootsie Roll will weigh between 3.25 and 3.45 grams

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:

[tex]Z=\dfrac{X-\mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

The Empirical Rule is also used to solve this question. It states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

In this problem, we have that:

[tex]\mu=3.30,\sigma=0.13[/tex]

(a) Within what weight range will the middle 95 percent of all miniature Tootsie Rolls fall?

By the Empirical Rule the weight range of the middle 95% of all miniature Tootsie Rolls fall within two standard deviations of the mean. So

3.30 - 2 x 0.13 = 3.04

3.30 + 2 x 0.13 = 3.56

The middle 95 percent of all miniature Tootsie Rolls will fall between 3.04 grams and 3.56 grams.

(b) What is the probability that a randomly chosen miniature Tootsie Roll will weigh more than 3.50 grams?

This probability is 1 subtracted by the p-value of Z when X = 3.50. So

[tex]Z=\dfrac{X-\mu}{\sigma}[/tex]

[tex]Z=\dfrac{3.50-3.30}{0.13}[/tex]

[tex]Z=1.54[/tex]

[tex]Z=1.54[/tex] has a p-value of 0.9382.

1 - 0.9382 = 0.0618

6.18% probability that a randomly chosen miniature Tootsie Roll will weigh more than 3.50 grams.

c) What is the probability that a randomly chosen miniature Tootsie Roll will weigh between 3.25 and 3.45 grams?

This is the p-value of Z when X = 3.45 subtracted by the p-value of Z when X = 3.25. So

X = 3.45

[tex]Z=\dfrac{X-\mu}{\sigma}[/tex]

[tex]Z=\dfrac{3.45-3.30}{0.13}[/tex]

[tex]Z=1.15[/tex]

[tex]Z=1.15[/tex] has a p-value of 0.8749.

X = 3.25

[tex]Z=\dfrac{X-\mu}{\sigma}[/tex]

[tex]Z=\dfrac{3.25-3.30}{0.13}[/tex]

[tex]Z=-0.38[/tex]

[tex]Z=-0.38[/tex] has a p-value of 0.3520

0.8749 - 0.3520 = 0.5229

52.29% probability that a randomly chosen miniature Tootsie Roll will weigh between 3.25 and 3.45 grams.

Answer: The missing dimension of the rhombus in the given figure is Height of rhombus h  h=A/b=90/15= 6cm. so missing dimension is h=6cm

What is Dimension ?

In general, dimension refers to the measurement or size of an object, space, or quantity along a particular axis or direction. In mathematics, dimension refers to the number of coordinates needed to specify a point in a space.

What is Rhombus ?

A rhombus is a type of quadrilateral (a four-sided polygon) in which all four sides are of equal length. It is a special case of a parallelogram in which the opposite sides are parallel to each other, and its opposite angles are equal.

In the given question,

area of rhombus is A=b*h  so it can be rewritten as  h=A/b by substituting values given in question we get h= 6cm

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The standard form of the equation of the ellipse is (x - 2)^2 / 4 + (y - 3)^2 / 7 = 1

To find the standard form of the equation of the ellipse, we first need to determine some of its properties.

The foci of the ellipse are given as (2, 0) and (2, 6). This tells us that the center of the ellipse is at the point (2, 3), which is the midpoint of the line segment connecting the foci.

The major axis of the ellipse is given as a length of 8. Since the major axis is the longest dimension of the ellipse, we can assume that the length of the major axis is 2a = 8, so a = 4.

Next, we need to determine the length of the minor axis. We know that the distance between the foci is 2c = 6, so c = 3. Since c is the distance from the center of the ellipse to each focus, we can use the Pythagorean theorem to find the length of the minor axis

b^2 = a^2 - c^2

b^2 = 4^2 - 3^2

b^2 = 7

b = sqrt(7)

Now we have all the information we need to write the standard form of the equation of the ellipse. The standard form is

(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1

where (h, k) is the center of the ellipse. Plugging in the values we found, we get

(x - 2)^2 / 4 + (y - 3)^2 / 7 = 1

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The answer to this question according to given information is 10. The intersection of the sets A and C, denoted as (A∩C), is the set of all elements present in both A and C.

Intersection is a term used to describe the common elements between two or more sets. A set is called as a collection of distinct objects, usually represented by a list, table, or diagram.  In other words, it is the overlap between the two sets. For example, the intersection of the set of even numbers and the set of prime numbers would be the set of all even prime numbers.

This is calculated by finding the common elements in the two sets. In this case, (A∩C) = {10, 13, 17, 23}.

Next, the intersection of (A∩C) and B, denoted as (A∩C)∩B, is the set of all elements which are common to all three sets. This is calculated by finding the common elements in (A∩C) and B. In this case, (A∩C)∩B = {10, 17}.

Finally, the number of elements in the set (A∩C)∩B is 10, since there are 10 elements in the set. To summarize, the number of elements in the set (A∩C)∩B is 10.

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The number of elements in the set (A∩C)∩B is 10. The intersection of the sets A and C, denoted as (A∩C), is the set of all elements present in both A and C.

Intersection is a term used to describe the common elements between two or more sets. A set is called as a collection of distinct objects, usually represented by a list, table, or diagram.  

This is calculated by finding the common elements in the two sets.

In this case, (A∩C) = {10, 13, 17, 23}.

Next, the intersection of (A∩C) and B, denoted as (A∩C)∩B, is the set of all elements which are common to all three sets.

This can be calculated by finding the common elements in (A∩C) and B.

In this case, (A∩C)∩B = {10, 17}.

Finally, the number of elements in the set (A∩C)∩B is 10, since there are 10 elements in the set.

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The length of JK is 18.333.

Since MN is parallel to JK, the angles formed by JLN and MLK are equal. Therefore, we can use the Triangle Proportionality Theorem, which states that if a line parallel to one side of a triangle divides the other two sides proportionally, then the triangles are similar.

Using the Triangle Proportionality Theorem, we can set up the following proportion:

[tex]$\frac{LK}{JL} = \frac{MN}{LN}$[/tex]

Therefore,

[tex]$\frac{7.2}{13.2} = \frac{10}{6}$[/tex]

We can cross-multiply to solve for JK:

[tex]$7.2 \cdot 6 = 13.2 \cdot 10$\\$43.2 = 132$\\$JK = \frac{132}{7.2} = 18.333$[/tex]

Therefore, the length of JK is 18.333.

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(a) The country will first experience shortages of food in approximately 26.6 years

(b) If the country doubled its initial food supply and maintained a constant rate of increase in the supply, shortages would still occur in approximately 38 years.

(c) If the country doubled the rate at which its food supply increases, in addition to doubling its initial food supply, shortages would still occur in approximately 55.4 years.

a. Let P(t) be the population of the country at time t (in years), and F(t) be the food supply of the country at time t.

We know that P(0) = 4 million, and P'(t) = 0.05P(t), which means that the population is increasing by 5% per year.

We also know that F(0) = 8 million, and F'(t) = 0.25 million, which means that the food supply is increasing by 0.25 million people per year.

When the food supply is just enough to feed the population, we have P(t) = F(t), so we can solve for t as follows:

4 million x (1 + 0.05)^t = 8 million + 0.25 million x t

[tex]4(1 + 0.05)^t = 8 + 0.25t\\\\t \approx 26.6 \ years[/tex]

b. If the country doubled its initial food supply, then F(0) = 16 million. We can use the same equation as before and solve for t:

4 million x  (1 + 0.05)^t = 16 million + 0.25 million x t

[tex]4(1 + 0.05)^t = 16 + 0.25t\\\\t \approx 38 \ years[/tex]

c. If the country doubled the rate at which its food supply increases and doubled its initial food supply, then we have F(0) = 16 million and F'(t) = 0.5 million. Using the same equation as before, we get:

4 million x  (1 + 0.05)^t = 32 million + 0.5 million x t

[tex]4(1 + 0.05)^t = 32 + 0.5t\\\\t \approx 55.4 \ years[/tex]

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No, the graph of tan 0, -1r/2 < 0 < 1r/2, is not asymptotic to the lines x = i and x = -i.

An asymptote is a line that a graph approaches but never crosses. The graph of tan 0, -1r/2 < 0 < 1r/2, has a period of π, meaning it repeats after every π, and will never cross the lines x = i and x = -i. This can be seen in the equation y = tan 0, where the x-values of -1r/2 and 1r/2 are replaced with the x-values of i and -i. The equation would be y = tan(i) and y = tan(-i), and the graphs of these equations would not be asymptotic to the lines x = i and x = -i.No, the graph of tan 0, -1r/2 < 0 < 1r/2, is not asymptotic to the lines x = i and x = -i.

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Answer:

[tex]p(x)=(x+5)(x-3)(x+4)[/tex]

Step-by-step explanation:

Given : [tex]p(x)=x^3+6x^2-7x-60[/tex]

Solution :

Part A:

First find the potential roots of p(x) using rational root theorem;

So, [tex]\text{Possible roots = }\pm\frac{\text{factors of constant term}}{\text{factors of leading coefficient}}[/tex]

Since constant term = -60

Leading coefficient = 1

[tex]\text{Possible roots = }\pm\frac{\text{factors of 60}}{\text{factors of 1}}[/tex]

[tex]\text{Possible roots = }\pm\frac{\text{1,2,3,4,5,6,10,12,15,20,60}}{\text{1}}[/tex]

Thus the possible roots are [tex]\pm1,\pm2,\pm3,\pm4,\pm5,\pm6,\pm10,\pm12,\pm15,\pm20,\pm60[/tex]

Thus from the given options the correct answers are -10, -5, 3, 15

Now For Part B we will use synthetic division

Out of the possible roots we will use the root which gives remainder 0 in synthetic division :

Since we can see in the figure With -5 we are getting 0 remainder.

Refer the attached figure

We have completed the table and have obtained the following resulting coefficients: 1 , 1,−12,0.  All the coefficients except the last one are the coefficients of the quotient, the last coefficient is the remainder.

Thus the quotient is

And remainder is 0 .

So to get the other two factors of the given polynomial we will solve the quotient by middle term splitting

[tex]x^2+x-12=0[/tex]

[tex]x^2+4x-3x-12=0[/tex]

[tex]x(x+4)-3(x+4)=0[/tex]

[tex](x-3)(x+4)=0[/tex]

Thus x - 3 and x + 4 are the other two factors

So, p(x)=(x+5)(x-3)(x+4)

Answer:

8

Step-by-step explanation:

Each side of the. decagon is equal so each side is 8

The integral expression for the volume of the solid formed by revolving the region bounded by the graphs of y = x₂ - 3x and y = x about the horizontal line y = 6 is  2πx(6 - x² + 3x)dx, which is integrated from x=0 to x=3, which gives us   81π/2.

To find the integral expression for the volume of the solid formed by revolving the region bounded by the graphs of y=x² - 3x and y=x about the horizontal line y=6, we can use the method of cylindrical shells.

First, we need to find the limits of integration, The graphs of y = x² - 3x  and y=x intersect at x=0 and x=3. Therefore, we integrate from x=0 to x=3.

Next, we consider a vertical strip of width dx at a distance x from the y- boxes. the height of the strip is the difference between the height of the curve y=  x² - 3x and the line y=6, which is   6 - (x² - 3x) = 6 - x² + 3x. the circumference of the shell is 2π times the distance x from the y-axis, and the thickness of the shell is dx. the volume of the shell is the product of the height, circumference, and thickness which is

dV = 2πx(6 - x² + 3x)dx

To find the total volume, we integrate this expression from x=0 to x=3.

V = ∫₀³ 2πx(6 - x² + 3x)dx, after simplifying the integrand we get :

V = 2π ∫₀³ (6x - x³ + 3x²)dx, integrating term by term we get :

V = 2π [(3x²/2) - (x⁴/4) + (x^3)] from 0 to 3, now  evaluation at the limits of integration we get:

V = 2π [(3(3)²/2) - ((3)⁴/4) + (3)³] - 2π [(0)^2/2 - ((0)⁴/4) + (0)^3]=  2π [(27/2) - (27/4) + 27] - 0 = 81π/2

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