Prove that the sum of the squares of the diagonals of a parallelogram is equal to the sum of squares of its sides.

The volume V of the resulting solid is [tex]$\frac{5000 \pi}{3}[/tex]

As per the data given:

The region bounded by the given curves is rotated about the specified axis.

Here we have to determine the volume V of the resulting solid by any method.

The given equations are x = [tex](y -6)^2[/tex], x = 25; about y = 1.

To find the volume of resulting solid we are using cylinder method

To derive the above solution, we need to use the old fashioned quadratic method.

Substitute the values of x then we get

25 = [tex](y-6)^2[/tex]

[tex]\rightarrow 5^2=(y-6)^2[/tex]

y - 6 = 5

y = 6 + 5

y = 11

So, 1 ≤ y ≤ 11

Graph attached at end of the solution

Using cylinder method and integrate along y -axis from y = 1 to y = 11

Volume [tex]}=\int_a^b 2 \pi r h d y[/tex]

Volume  [tex]& =2 \pi \int_1^{11}(y-1)\left(25-(y-6)^2\right) d y[/tex]

Volume  [tex]=2 \pi \int(y-1)\left(25-y^2-36+12 y\right) d y \\[/tex]

Volume [tex]& =2 \pi \int_1^{11}\left(-y^3+13 y^2-23 y+11\right) d y[/tex]

Volume  [tex]=2 \pi\left[-\frac{y}{4}+\frac{13 y}{3}-\frac{23 y}{2}+11 y\right]_1^{11} \\[/tex]

Volume [tex]& =2 \pi \cdot\left(\frac{10043}{12}-\frac{43}{12}\right)=2 \pi\left(\frac{10000}{12}\right)=\frac{5000\pi }{3}[/tex]

Thus the volume of resulting solid is:

[tex]$}=\frac{5000 \pi}{3}[/tex]

Therefore the volume is [tex]$\frac{5000 \pi}{3}[/tex]

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Using the figure the true trigonometry statements are

A.  (sin A)² + (cos A)² = 1

C. 1/(cos A)² - (tan A)² = 1

D. (sin A)² + (cos A)² + (tan A)²  = 1/(cos A)²

A. (sin A)² + (cos A)² = 1 this is a fundamental trigonometry identity

say angle A = 30

(sin 30)² + (cos 30)² = 1

C. 1/(cos A)² - (tan A)² = 1

1/(cos A)²= 1 + (tan A)²

1/(cos A)²= 1 + (sin A)² / (cos A)²

1/(cos A)²= ((cos A)² + (sin A)²) / (cos A)²

and (cos A)² + (sin A)² = 1, hence

1/(cos A)²= 1/(cos A)² this is true

D. (sin A)² + (cos A)² + (tan A)²  = 1/(cos A)²

recall (sin A)² + (cos A)² = 1

1 + (tan A)²  = 1/(cos A)² same as line 2 in C hence this is true

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For the first statement, the likely explanatory variable is the miles per gallon and the likely response variable is the weight.

For the second statement, the likely explanatory variable is the weight and the likely response variable is the miles per gallon.

In statistical analysis, an explanatory variable is a variable that is presumed to have an effect on another variable. It is also known as an independent variable. On the other hand, a response variable is the variable that is presumed to be affected by the explanatory variable. It is also known as a dependent variable.

In the first statement, it is stated that the explanatory variable is the miles per gallon, which means that we are looking at how the fuel efficiency of a vehicle affects its weight. Therefore, the weight is the likely response variable, as it is presumed to be affected by the miles per gallon.

In the second statement, it is stated that the explanatory variable is the weight, which means that we are looking at how the weight of a vehicle affects its fuel efficiency. Therefore, the miles per gallon is the likely response variable, as it is presumed to be affected by the weight.

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Answer:

B. Vertex

Step-by-step explanation:

The midpoint between the two x-intercepts of a parabola will provide the x-coordinate of the vertex. Line of symmetry is drawn along this particular  x-coordinate. This x-coordinate is substituted into the equation of the graph to find its corresponding y-coordinate.

(a) Revolving about the x-axis, the volume of the solid is 3Pi

(b) Revolving about the y-axis, the volume of the solid is 18 Pi

(c) Revolving about the line x=3, the volume of the solid is 0

(d) Revolving about the line x=6, the volume of the solid will be 32/3 Pi

The region bounded by y=x, y=0, and x=3 is a triangle in the first quadrant.

To find the volume of the solid generated by revolving this region about an axis, we can use the disk or washer method.

(a) Revolving about the x-axis:

Each cross-section of the solid perpendicular to the x-axis is a disk with radius x and thickness dx.

The volume of each disk is [tex]\pi x^2 dx[/tex].

The limits of integration are 0 and 3, the x-coordinates of the intersection points of the two curves.

Hence, the volume of the solid is

[tex]V &= \int_0^3 \pi x^2 dx\\\\\&= \left[\frac{\pi}{3}x^3\right]_0^3\\\\\&= \frac{9\pi}{3}\\\\\&= 3\pi.[/tex]

(b) Revolving about the y-axis:

Each cross-section of the solid perpendicular to the y-axis is a washer with outer radius 3 and inner radius x, and thickness dx.

The volume of each washer is [tex]\pi(3^2-x^2)dx.[/tex]

The limits of integration are 0 and 3.

Hence, the volume of the solid is

[tex]V &= \int_0^3 \pi(3^2-x^2)dx\\\\\&= \left[\pi(9x-\frac{x^3}{3})\right]_0^3\\\\\&= \pi(27-\frac{27}{3})\\\\\&= 18\pi.[/tex]

(c) Revolving about the line x=3:

Each cross-section of the solid perpendicular to the line x=3 is a washer with outer radius 3-x and inner radius 3-x-|x|, and thickness dx.

The volume of each washer is[tex]\pi((3-x)^2-(3-x-|x|)^2)dx[/tex].

The limits of integration are -3 and 3.

Hence, the volume of the solid is

[tex]V &= \int_{-3}^3 \pi((3-x)^2-(3-x-|x|)^2)dx\\\\\&= \left[8\pi\int_0^3 (3-x-|x|)dx\right]\\\\\&= 8\pi\int_0^3 (3-2x)dx\\\\\&= 8\pi\left[3x-x^2\right]_0^3\\\\\&= 8\pi(9-9)\\\\\&= 0.\\\\[/tex]

(d) Revolving about the line x=6:

Each cross-section of the solid perpendicular to the line x=6 is a washer with outer radius |3-x| and inner radius |x-6|, and thickness dx.

The volume of each washer is[tex]\pi(|3-x|^2-|x-6|^2)dx.[/tex]

The limits of integration are 0 and 3.

Hence, the volume of the solid is

[tex]V &= \int_0^3 \pi(|3-x|^2-|x-6|^2)dx\\\\\&= \left[\frac{32\pi}{3}\right]\\\\\&= \frac{32}{3}\pi.[/tex]

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Answer:

2/3 x 2/3 as a fraction is 4/9.

Step-by-step explanation:

We are performing simple fractional multiplication to get the answer, i.e., 2 x 2 = 4. The numerator of the fraction is multiplied with the numerator of the other, i.e., 3 x 3 = 9.

Answer:

4/21

Step-by-step explanation:

Just line them up side by side, then multiply 2×2 and 3×7. Hope this helps

Answer:

6 combinations

Step-by-step explanation:

3 pairs of Jeans  & 2 shirts

3 x 2 = 6

6 combinations

Answer:

The x-intercept is 27/8, or 3.375.

Step-by-step explanation:

0 = -8/3x + 9

-8/3x = -9

-9 * -3/8 = 27/8

x = 27/8

Answer: 6 days

120 miles divided into 30 days is 4 miles per day.

24 miles divided by 4 miles is 6 , which means the bear spent 6 days walking along Kissimmee River.

I hope this helped & Good Luck <3 !!

a. Yes, Mia Hamm has an absolute advantage in both scoring goals and saving shots as a goalie,

b. The opportunity cost of scoring 1 goal for Mia is 0.6 shots

c. The opportunity cost of saving 1 shot as a goalie for Mia is 1.67 goals . The opportunity cost of saving 1 shot as a goalie for Alyssa is 0.17 goals scored.

d. Mia has the comparative advantage in scoring goals, since her opportunity cost of scoring 1 goal is lower than Alyssa's. Alyssa has the comparative advantage in saving shots as a goalie, since her opportunity cost of saving 1 shot is lower than Mia's

b.  The opportunity cost of scoring 1 goal for Mia is 0.6 shots saved as a goalie  which is calculated as

30 shots/50 goals = 0.6).

The opportunity cost of scoring 1 goal for Alyssa is 6 shots saved as a goalie  which is calculate as

opportunity cost 120 shots/20 goals = 6.

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The generalization suggested by these results is that as the number of contrasts increases, the multiples decrease.

a. For each of the g numbers of pairwise comparisons in the family (2, 5, and 15), the T, S, and B multiples can be calculated using the following formulas:

[tex]T = t α/2 (g-1)S = t α/2 (g-1) √2/(n-1)B = t α/2 (g-1) √(2/n)[/tex]

Where tα/2 is the t-value associated with the given family confidence coefficient (in this case, 90%) and g is the number of pairwise comparisons.

For g=2, the T, S, and B multiples are 3.078, 2.179, and 2.500 respectively. For g=5, the multiples are 2.571, 1.711, and 2.001 respectively. For g=15, the multiples are 2.228, 1.429, and 1.641 respectively.

The generalization suggested by these results is that as the number of pairwise comparisons increases, the multiples decrease.

b. For each of the g numbers of contrasts in the family (2, 5, and 15), the S and B multiples can be calculated using the following formulas:

[tex]S = t α/2 (g-1) √2/nB = t α/2 (g-1) √(2/n)[/tex]

Where tα/2 is the t-value associated with the given family confidence coefficient (in this case, 90%) and g is the number of contrasts.

For g=2, the S and B multiples are 2.179 and 2.500 respectively. For g=5, the multiples are 1.711 and 2.001 respectively. For g=15, the multiples are 1.429 and 1.641 respectively.

The generalization suggested by these results is that as the number of pairwise comparisons or contrasts increases, the multiples decrease.

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The statement true for A, C and E is the second statement r → p ∨ q.

Truth table is a table which describes the truth value of a complex statement regarding to the truth value of single statements.

Given is a truth table.

Consider r → p ∨ q, for the statements A, C and E.

For A, p is true and q is true, then p ∨ q is true. So r is true.

For C, p is true and q is false, then p ∨ q is true. So r is true.

For E, p is false and q is true, then p ∨ q is true. So r is true.

Hence the correct statement is r → p ∨ q.

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Answer:

14.5

Step-by-step explanation:

2.9(5)=14.5

Answer:

14.5p

here,

lets start off by looking at the full scale problem. We already know that each crate can hold 2.9 apples at a time. We also know that we have 5 crates in total.

hence,

we would multiply [tex]5(2.9)=14.5[/tex]  or  [tex]5x2.9=14.5[/tex]

The current population of Glen Cove is 5,000 people and in the long-run, the population of Glen Cove will approach 35,000 people.

A corporation may be a corporate entity whose shareholders elect a board of directors to supervise its operations. the dissimilarity between the business and an organization is that an organization may be a sort of business that's suitable for smaller businesses and organizations, whereas a corporation may be a kind of business that's appropriate for larger enterprises and entities.

The population in Glen Clove's population that is t years from now will be

P(t) = [tex]\frac{35t^2 + 125t + 200}{t^2 + 4t + 40}[/tex]

The current population can be calculated by substituting t = 0 in P(t).

So,

P(0) = [tex]\frac{35(0)^2 + 125(0) + 200}{(0)^2 + 4(0) + 40}[/tex]

P(0) = 5

Thus, the current population of Glen Clove is 5 thousand.

(b)For the long run, t tends to infinity. So, evaluate the infinite limit for the given function as follows:

[tex]\lim_{t \to \infty} P(t)= \lim_{t \to \infty}P(t)\frac{35t^2 + 125t + 200}{t^2 + 4t + 40}[/tex]

                     =[tex]\lim_{t \to \infty}\frac{t^2(35+\frac{125}{t} +\frac{200}{t} )}{t^2(1+\frac{4}{t} +\frac{40}{t^2} )}[/tex]

                    = [tex]\lim_{t \to \infty}\frac{(35+\frac{125}{t} +\frac{200}{t} )}{1+(\frac{4}{t} +\frac{40}{t^2} )}[/tex]

Now, solve further by putting the limits as follows:

As t approaches infinity, the terms in the numerator approach 0, while the terms in the denominator approach 1, so the limit of P(t) as t approaches infinity is:

lim t -> infinity P(t) = 35 / 1 = 35

So, in the long-run, the population of Glen Cove will approach 35,000 people.

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The solution is, the line of reflection is y = 1.

When reflecting a figure in a line or in a point, the image is congruent to the preimage. A reflection maps every point of a figure to an image across a fixed line. The fixed line is called the line of reflection.

here, we have,

The given figure shows two triangles reflected horizontally.}

We can find the line reflection because it would be an horizontal line half way. Remember that horizontal lines have the form y = k, where k = 1 in this case,

Hence, the line of reflection is y = 1.

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The statement that is true about the total cost of the carpet and installation is: The total cost of the carpet and installation is $633.

What is Algebraic expression ?

Algebraic expression can be defined as combination of variables and constants.

To determine the total cost of the carpet and installation, we first need to calculate the area of Wendy's bedroom floor, which is:

Area = Length x Width

Area = 15 ft x 12 ft

Area = 180 square feet

We can then convert the area to square yards by dividing by 9:

Area in square yards = 180 sq ft / 9

Area in square yards = 20 sq yd

The cost of the carpet is $27.50 per square yard, so the cost of the carpet alone is:

Carpet cost = 20 sq yd x $27.50/sq yd

Carpet cost = $550

The installation cost is $4.15 per square yard, so the cost of the installation alone is:

Installation cost = 20 sq yd x $4.15/sq yd

Installation cost = $83

To find the total cost, we need to add the cost of the carpet and the installation

Total cost = Carpet cost + Installation cost

Total cost = $550 + $83

Total cost = $633

Therefore, the statement that is true about the total cost of the carpet and installation is: The total cost of the carpet and installation is $633.

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a) C(x) = 0.02x + 2200, b) A straight line with positive slope, c) Slope is 0.02, represents cost per chair, d) Y-intercept is 2200, represents fixed cost.

a) The cost C(x) can be expressed as a linear function of the number of chairs x produced. We can solve for the slope and y-intercept using the two points (100, 2200) and (300, 4800). First, calculate the slope of the line using the formula (y2 - y1)/(x2 - x1). This gives us (4800 - 2200)/(300 - 100) = 0.02. Then, we can use the slope and one of the points to solve for the y-intercept using the equation y = mx + b, where m is the slope and b is the y-intercept. Substituting the values from the first point (100, 2200) into the equation gives us 2200 = 0.02(100) + b, so the y-intercept is 2200. Therefore, the cost function can be expressed as C(x) = 0.02x + 2200.

b) The graph of this function is a straight line with a positive slope, as shown in the diagram.

c) The slope of the graph is 0.02, which represents the cost per chair.

d) The y-intercept is 2200, which represents the fixed cost of manufacturing the chairs.

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a. The amplitude is 12m, the midline is 13m, and the period of h (t) is 16 minutes.

a. The amplitude of the height function h(t) is 12 meters (24 meters diameter / 2).

The midline of the height function is 12 meters (24 meters diameter / 2) + 1 meter (height of the platform).

The period of the height function is the time it takes for the Ferris wheel to complete one full cycle, which is 16 minutes.

b. The height function h(t) can be modeled as a sinusoidal function, where h(t) = 12 cos (2πt/16) + 13.

The cosine function models the cyclical change in height as the Ferris wheel turns.

The 2π in the argument of the cosine function represents the full revolution of the Ferris wheel, and the 16 in the argument of the cosine function represents the time it takes for the Ferris wheel to complete one revolution.

The 13 at the end of the equation is the midline of the height function, which represents the average height of the person above the ground.

c. To find the height of a person after 52 minutes, we substitute t = 52 into the height function h(t) = 12 cos (2πt/16) + 13:

h(52) = 12 cos (2π x 52/16) + 13

h(52) = 12 cos (13π) + 13

h(52) = 12(-1) + 13

h(52) = 1 meter

So, a person would be 1 meter above the ground after 52 minutes.

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Factoring the expression 16x+ 8y using GCF would be, 8(2x + 1).

The GCF of two or more than two numbers is the highest number that divides the given two numbers completely.

We also know that distributive property states a(b + c) = ab + ac.

Given, An expression 16x+ 8y.

Now, The HCF of 16 and 8 is 8, Because 8 is the highest number that divides 16 and 8 completely.

Therefore, The factored form of the expression 16x+ 8y would be,

8(2x + 1).

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The value of the scale factor of dilation (k) is 5/3 and the value of x is 21

The variable k is the scale factor of dilation of the triangles

This is calculated as

k = Image triangle/Pre-image triangle

Substitute the known values in the above equation, so, we have the following representation

k = 15/9

So, we have

k = 5/3

This is calculated as

x * 5/3 = 35

So, we have

x = 3 * 35/5

Evaluate

x = 21

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The expected number of defects per yard is 1.

The unconditional probability distribution for Y, the number of defects per yard in a certain fabric, is P(y) = - (1/2)^y+1 y = 0, 1, 2, ....

To calculate the expected number of defects per yard, we use the formula E(Y) = ΣyP(y), where y is the number of defects per yard and P(y) is the probability of that number of defects per yard.

We can then substitute the values for P(y) into the formula and sum for all values of y from 0 to ∞. This yields E(Y) = Σy(-(1/2)^y+1) = 1. Thus, the expected number of defects per yard is 1.

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Answer:

$33.52

Step-by-step explanation:

7.89+7.89+15= 30.78

30.78x 0.089=2.73942 rounded off to the nearest hundredth which is 2.74

30.78+2.74= $33.52

Step-by-step explanation:

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The only solution to the given logarithmic equation is x = 4.

An example of a logarithmic function is one that expresses the connection between two values as the logarithm of one quantity to a certain base.

A logarithmic function's basic form is:

f(x)=log base b (x)

where x is the function's input, b is the logarithm's base, and f(x) is the function's value.

We can solve the logarithmic equation step by step as follows:

log₂[log₂ (√4x)] = 1

First, we simplify the expression inside the logarithm:

log₂[log₂ (√4x)] = 1

log₂[log₂ (2√x)] = 1

Next, we use the definition of logarithms to rewrite the equation in exponential form:

log₂ (2√x) = 2¹ = 2

2√x = 2² = 4

√x = 2

x = 4

Thus, the solution is x=4.

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Answer:

b. 2/3

Step-by-step explanation:

8 = 4 x 2
√8 = √4 x √2

√4 = 2

√8 = 2 x √2 = 2√2

18 = 9 x 2

√18 = √9 x √2

√9 = 3

√18  = 3 x √2 =3√2

√(8/18) = 2√2 / 3√2 = 2/3

The equation that represents the relationship between the number of miles to tread thickness will be y = - 0.12x + 9.

Let the equation of the line pass through (x₁, y₁) and (x₂, y₂). Then the equation of the line is given as,

[tex]\rm (y - y_2) = \left (\dfrac{y_2 - y_1}{x_2 - x_1} \right ) (x - x_2)[/tex]

Let 'x' be the number of miles and 'y' be the tread thickness. Then the two points are (15, 7.2) and (35, 4.8). Then the equation is given as,

(y - 7.2) = [(7.2 - 4.8) / (15 - 35)] (x - 15)

y - 7.2 = -0.12(x - 15)

y - 7.2 = - 0.12x + 1.8

y = - 0.12x + 9

The equation that represents the relationship between the number of miles to tread thickness will be y = - 0.12x + 9.

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As per compound interest, the account balance at the end of 12 years is approximately $17,217.

Compound interest is an important concept in finance and investing, and it plays a critical role in understanding how your money can grow over time. When you invest money, your initial investment earns interest, and over time, that interest can also earn interest, resulting in exponential growth.

In the context of this question, the investment into a college fund is represented by the equation

=> [tex]A = 12,000e^{0.023*12}[/tex]

where A is the amount of money in the account at the end of the time given. Let's break down the different components of this equation and see how they relate to compound interest.

First, we have the initial investment, which is not explicitly given in the equation, but we can infer that it is 12,000. This is the amount of money that is initially invested into the college fund.

Next, we have the interest rate, which is 0.023.

Finally, we have the amount of time that the money is invested, which is 12 years in this case. The longer the money is invested, the more time there is for compound interest to work its magic and grow the account balance.

Putting all of these components together, we can use the equation

=> [tex]A = 12,000e^{0.023*12}[/tex]

to calculate the amount of money in the college fund at the end of 12 years.

By plugging in the values for the initial investment, interest rate, and time, we can see that the account balance at the end of 12 years is approximately $17,217.

This represents the power of compound interest and how it can turn a modest initial investment into a substantial sum over time.

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Answer:

2

Step-by-step explanation:

ab - 10. this means 10 is less than the product of a and b

The residual of a customer with a height of 155 cm who rents a bike with a 51 cm frame is -1.

The discrepancy between a response variable's observed value and its expected value as predicted by the regression line.

We have the Equation: y'=1/3x + 1/3.

The regression equation is given as:

y'=(1/3)x + (1/3)

Since the height is given as 155cm, x=155 cm

The predicted frame size,

y'=(1/3)x + (1/3)

y'=(1/3) × 155+ (1/3)

  = 51 2/3 + 1/3

  = 52

So, Residual = Observed y- predicted y

                     = 51-52

                     = -1

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Answer:

Step-by-step explanation:

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The length of x is 7.

In geometry, a triangle is a three-sided polygon made up of three line segments that join at three points called vertices. Triangles are one of the most basic and important shapes in geometry, and they are used in many areas of mathematics, science, and engineering.

Using the property of a median, we know that BD divides AC into two equal parts. Let's call the length of CD as y.

So, we have:

AC = AD + CD

x = 3 + y

Using the Pythagorean Theorem, we can find the length of BD:

BD² = AB² + AD² - 2(AB)(AD)(cos(B))

BD² = 4² + 3² - 2(4)(3)(cos(B))

BD² = 25 - 24cos(B)

BD = √(25 - 24cos(B))

Since BD is a median, we know that it divides BC into two equal parts. Let's call the length of BD as z.

So, we have:

BC = 2z

8 = 2z

z = 4

Now, using the property of medians again, we know that:

4² + y² = z²

16 + y² = 16

y² = 0

y = 0

Therefore, CD is also 4, and:

x = 3 + y

x = 3 + 4

x = 7

So, the length of AC is 7.

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