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The average value of 3t^3-t^2 over the interval -1<=t<=2 is 11/4

Mean Value Theorem states if a function f(x) is continuous on the interval [a,b] and differentiable on (a,b), there is at least one value c in the interval (a<c<b) such that:

f '(c)=f(b)-f(a)/b-a

Average Value:

The average value of a function of a variable over a range represents the height of a rectangle of the same area as that defined by the function under the curve. This value can be calculated using a formula:

favg=1/b−a∫f(t)dt

consider f(x)=3t³-t² and limits are -1 to 2

Now, substitute these values in the above formula:

favg=1/2-(1)∫3t³-t²dt

favg=1/3∫(3/4t⁴-1/3t³)|₋₁²

favg=(12-8/3)-(3/4+1/3)

By solving the equation we get

favg=11/4

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Answer:

4,4

Step-by-step explanation:

44x1/10

Answer:

5 & -4

4 & -3

3 & -2

2 & -1

Step-by-step explanation:

5 + (-4) = 1

4 + (-3) = 1

3 + (-2) = 1

2 + (-1) = 1

The surface area of the capsule is calculated by adding the surface area of the cylinder and the surface area of the hemispherical bowl.

A hemispherical bowl is a type of bowl with a rounded bottom that is shaped like half of a sphere. It is often used in laboratories, kitchens, and dining rooms. The bowl’s curved surface prevents the contents from spilling, which makes it an ideal choice for mixing, swirling, and scooping. Additionally, the bowl’s shape makes it look aesthetically pleasing, so it can be used as a decorative piece as well.

The surface area of the cylinder = 2πr2 + 2πrh

Where r is the radius of the cylinder and h is the height of the cylinder.

The radius of the cylinder = 12 cm

The height of the cylinder = 16 cm

Therefore, the surface area of the cylinder = 2π(12)2 + 2π(12)(16) = 1536π cm2

The surface area of the hemispherical bowl = 2πr2

Where r is the radius of the hemispherical bowl.

The radius of the hemispherical bowl = 12 cm

Therefore, the surface area of the hemispherical bowl = 2π(12)2 = 144π cm2

The total surface area of the capsule = 1536π cm2 + 144π cm2 = 1680π cm2

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Answer:

Step-by-step explanation: Yes, they can match in 2, 5, and/or 10 equal rows. This is because the number of items in the set can be divided evenly by 2, 5, and 10. For example, if there are 20 items in the set, it can be divided into 2 rows of 10, 5 rows of 4, or 10 rows of 2.

Answer:

-3

Step-by-step explanation:

rise over run method

3 up and 1 to the left which makes the slope negative

(A) f(-1) = 1

(B) The range of the function f(x) is [-4, ∞).

What is the range of the function?

In mathematics, a function's range can refer to one of two closely related ideas: Its codomain is a function. The picture of the action Given two sets X and Y, a binary relation f between them is a function if and only if there is precisely one y in Y such that f connects x to y for every x in X.

For the given function we have to find the value of f(-1) and to find the range of the function.

From graph we can see that,

f(-1) = 1

The range of the function is the interval in which function cover the x - axis.

The range of the function is [-4, ∞).

Hence,

(A) f(-1) = 1

(B) The range of the function f(x) is [-4, ∞).

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The range of the given function y = 3x - 2 for the domain (-1, 2, 4) is

A function is a relation from a set A to a set B where the elements in set A only maps to one and only one image in set B. No elements in set A has more than one image in set B.

Domain of the function is the values of x for which the function is defined.

Range of the function is the value of the function when we put the values in the domain.

Given function,

y = 3x - 2

Given the domain (-1, 2, 4)

x = -1, then y = (3 × -1) - 2 = -5

x = 2, then y = (3 × 2) - 2 = 4

x = 4, then y = (3 × 4) - 2 = 10

Range is (-5, 4, 10).

Hence the range of the function is (-5, 4, 10).

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Answer:

a)  (-∞, -1) ∪ (-1, ∞)

b)  (-∞, -4) ∪ (-4, ∞)

c)  Vertical asymptote: x = -1

    Horizontal asymptote:  y = -4

Step-by-step explanation:

Given function:

[tex]f(x)=\dfrac{1}{x+1}-4[/tex]

The domain of a function is the set of all possible input values (x-values).

When the denominator of a rational function is zero, the function is undefined.  

Rewrite the function as one fraction:

[tex]\implies f(x)=\dfrac{1}{x+1}-\dfrac{4(x+1)}{x+1}[/tex]

[tex]\implies f(x)=\dfrac{1-4(x+1)}{x+1}[/tex]

[tex]\implies f(x)=\dfrac{-4x-3}{x+1}[/tex]

Set the denominator to zero and solve for x:

[tex]\implies x+1=0[/tex]

[tex]\implies x=-1[/tex]

Therefore, the given function is undefined when x = -1, so its domain in interval notation is:

The range of a function is the set of all possible output values (y-values).

As the domain is restricted to (-∞, -1) ∪ (-1, ∞), the range is also restricted.

To find the range of a rational function, first solve the equation for x:

[tex]\implies y=\dfrac{1}{x+1}-4[/tex]

[tex]\implies y+4=\dfrac{1}{x+1}[/tex]

[tex]\implies (y+4)(x+1)=1[/tex]

[tex]\implies x+1=\dfrac{1}{y+4}[/tex]

[tex]\implies x=\dfrac{1}{y+4}-1[/tex]

[tex]\implies x=\dfrac{1-(y+4)}{y+4}[/tex]

Set the denominator of the resultant equation ≠ 0 and solve for y:

[tex]\implies y+4 \neq 0[/tex]

[tex]\implies y \neq -4[/tex]

Therefore, the range is the set of all real numbers other than y = -4:

A vertical asymptote occurs at the x-value(s) that make the denominator of a rational function zero.

Therefore, there is a vertical asymptote at x = -1.

As the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the result of dividing the highest degree term of the numerator by the highest degree term of the denominator.

[tex]\implies f(x)=\dfrac{-4x-3}{x+1}[/tex]

Therefore, there is a horizontal asymptote at:

There are no slant asymptotes as there is a horizontal asymptote.

On solving the provided question we can say that the value of equation here will be as duration is 0<t<5 seconds.

An equation is a mathematical formula that connects two assertions using the equal sign (=) to denote equivalence. In algebra, an equation is a mathematical statement that establishes the equivalence of two mathematical expressions

Information given in the query:

Rooftop height is 70 feet.

Speed at start: 80 feet per second

s(t ) = - 16t² + 80t + 70

s(t) in feet here

now,

for the balls to touch the rooftop

[tex]s(t ) = - 16t^2+ 80t + 70 > 70\\ - 16t^2 + 80t + 70 > 70\\ - 16t^2 + 80t > 70 - 70\\ - 16t^2 + 80t > 0\\ 16t^2 - 80t < 0\\t( 16t - 80 ) < 0\\t < 0 and 16t - 80 < 0 or 16t < 80 or t < 5\\[/tex]

Hence,

since time cannot be negative, the time period is 0 t 5 seconds.

0<t<5

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The events Grade 5 and Science"  are independent events,  P(Grade 5 ∩ Science) = P(Grade 5) ⋅ P(Science) .Option B is the correct option.

What is frequency?

The frequency (f) of a specific value is the number of occurrences of the value in the data. The frequency distribution of a variable is the collection of all possible values and the frequencies associated with these values. Frequency distributions are represented graphically as frequency tables or charts.

Given table is

                   English      Science   Total

Grade 5           24              36           60

Grade 8           16               24           40

Total                40              60           100

Total students is 100.

Total student in grade 5 is 60.

The number of students who prefer science is 60.

The number of students who are in grade 5 and prefer science is 36.

The probability of grade 5 is P(grade 5) = 60/100 =0.6

The probability of science is P(science) = 60/100 = 0.6

The probability of Grade 5 ∩ Science is P(Grade 5 ∩ Science) = 36/100 =0.36.

Thus P(Grade 5 ∩ Science) =P(Grade 5) ⋅ P(Science).

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Two positive numbers whose product is 192 and whose sum is a minimum are; 8√3 and 8√3.

The correct option is (D).

Minima is the minimum value of a function in given domain.

First Order Derivative Test

Let f be the function defined in an open interval I and f be continuous at critical point c in I such that f’(c) = 0.

If f’(x) changes sign from negative to positive as x increases through point c, then c is the point of local minima. And the f(c) is the minimum value.

Given,

Product of two positive numbers = 192

Sum should be minimum

Let x be the first positive number and y be the other positive number. So, the equation would be

x . y = 192

⇒ y = 192/x       --------(a)

Let S be the sum of the two positive numbers.

S = x + y

substituting value of y in terms of x from equation(a)

S = x + 192/x

⇒ S = x + 192 . x⁻¹

Differentiating with respect to x

S'(x) = 1 + (-1)192x⁻²

S'(x) = 1 - 192/x²

S'(x) = (x² - 192)/x²

To determine the minimum S, equating first derivative of S with respect to x to zero.

S'(x) = 0

(x²-192)/x² = 0

x² - 192 = 0

x² = 192

x = ±√192

Rejecting negative root, since the numbers are positive

x = √192

Substituting value of x in equation (a)

y = 192/√192

y = √192

x and y can be written as

x = √192 = 8√3

y = √192 = 8√3

Hence , 8√3  and 8√3 are the two positive numbers whose product is 192 and sum is a minimum.

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Here, on solving the provide question to us, we got to know that - equation  210 feet represents  = 825/4

Since equations are essentially questions, efforts to systematically find answers to those questions have been the driving force behind the development of mathematics. From straightforward algebraic equations that just require addition or multiplication to differential equations, exponential equations that use exponential expressions, and integral equations, there are many different types of equations that range in complexity.

Here,

15 feet represents = 1/8  of an inch

1 feet represents = 1/8 X 15

The growth of mathematics has been fueled by efforts to methodically find answers to equations, which are basically questions. There are many distinct forms of equations that range in complexity, from simple algebraic equations that just need addition or multiplication to differential equations, exponential equations that employ exponential expressions, and integral equations 210 feet represents = 1/8 X 15 X 210 = 825/4

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The domain and range of the piecewise function is;

B) Domain: x < 2 or x ≥ 3.; range: y > -3

The piecewise function is given as;

f(x) = {-¹/₂x - 2, if x < 2

        {2x - 4, if x ≥ 3

The domain of a function is the set of all possible input values of the function and as such in this case the domain is x < 2 or x ≥ 3.

The range is defined as the set of all possible output values of the function and in this case, we have;

For -¹/₂x - 2; If x < 2

At x = 2; -¹/₂(2) - 2 = -3

Range is; y > -3

Similarly, for 2x - 4, if x ≥ 3

At x = 3; 2(3) - 4 = 2

Range is; y ≥ 2

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Answer:

Step-by-step explanation:

The vertex (h, k) is always in the format of "x-" so if we have a positive h value for the vertex, it would be written as "x-(h)" and if we have a negative h value for the vertex, it would be written as "x-(-h)" which is the same as "x+h". Since our sign inside the absolute value symbols is negative, the h value for the the vertex is 9. The k value is always what's outside the absolute value symbols, including the sign. Ours is a positive 7. Thus the vertex is (9, 7).

Answer: 40 dollars

Step-by-step explanation:

i hope the explanation is right:)

If he invites 5 friends for 40 bucks each then one friend that he knows will go is 40 dollars. So the only thing you have to do for this problem is multiply 40 by the number of friends which in this case i believe is 1.

Answer:

B,C

Step-by-step explanation:

B. All waves have both an amplitude and frequency.

C. All waves have a repeating pattern.

A statement A is incorrect, Waves are a disturbance that transfer energy through space without transferring matter.

B is true, All types of waves have an amplitude, which is the measure of the maximum displacement of a point on the wave from its rest position and a frequency, which is the number of complete oscillations of a point on the wave per unit of time.

C is also true, All types of waves have a repeating pattern, which is known as a waveform. The shape of the waveform determines the type of wave. For example, sine waves have a smooth, curved shape, while square waves have sharp, angular edges.

On solving the provided question we can say that the inequality equation can solved as  = [tex]\sqrt[4]{xyzw}[/tex]

An inequality in mathematics is a relationship between two expressions or values that is not equal. Thus, imbalance leads to inequality. An inequality creates the link between two values that are not equal in mathematics. Egality is distinct from inequality. When two values are not equal, most commonly use the not equal sign (). Different inequalities are used to contrast values, no matter how little or large. Many simple inequalities may be resolved by modifying the two sides until the variables are all that remain. But a number of things contribute to inequality: Negative values on both sides are divided or added. Trade off the left and right.

here,

the inequality equation can solved as

[tex]x^m+y^m+z^m+w^m/x^m+y^-m+z^-m+w^-m= (xyzw)^{m/2}[/tex]

[tex]u = x+y/2 \geq \sqrt{xy}[/tex]

v = z+w /2 [tex]\geq \sqrt{zw}[/tex]

u+v/2 = x+y+z+w+/4 [tex]\geq \sqrt{uw} \geq \sqrt{\sqrt{xy} \sqrt{zw} }[/tex]

= [tex]\sqrt[4]{xyzw}[/tex]

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Distance traveled after toll payment is 1.2 miles on highway 40.

The distance may be calculated using a curved route. Displacement measurements can only be made along straight lines. Distance is path-dependent, meaning it varies depending on the direction followed. Displacement simply depends on the body's beginning and ending positions; it is independent of the route.

Distance is the sum of an object's movements, regardless of direction. Distance may be defined as the amount of space an item has covered, regardless of its beginning or finishing position.

The size or extent of the displacement between two points is referred to as distance. Keep in mind that the distance between two points and the distance traveled between them are not the same. The entire length of the journey taken between two points is known as the distance traveled. Travel distance is not a vector.

Distance traveled before toll payment =3/5 miles on highway 40

Distance traveled after toll payment =2*3/5  = 6/5 =

1.2 miles on highway 40.

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Considering the coordinates of the preimage of line AB represented by A (6, -4), B (2, -8) and coordinates of the image of line A'B' represented by A' (9, -6), B'(3, -12) the dilation factor is 3/2

The transformation rule for dilation is as follows

(x, y) for a scale factor of r → (rx, ry)

following similar procedure for the given problem we have

A (6, -4) transformed to A' (9, -6)

B (2, -8) transformed to B' (3, -12)

hence 6r = 9, and -4r = -6

r = 9/6 = -6/-4

r = 3/2

The dilation factor of the transformation is 3/2

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The number of red counters in the bag is given as follows:

8.

A probability is obtained by the division of the number of desired outcomes by the number of total outcomes.

The probabilities for this problem are given as follows:

The sum of the probabilities is of one, hence the value of x is obtained as follows:

2x + x + 0.45 + 0.25 = 1

3x = 0.3

x = 0.1.

Hence the proportion of red is given as follows:

0.2.

There are 18 blue counters in the bag, which is of 0.45 of the total amount, hence the total amount is obtained as follows:

0.45n = 18

n = 18/0.45

n = 40.

Meaning that the number of red counters is given as follows:

0.2 x 40 = 8.

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Answer:

Copy the table into your notebook and include columns to find the endpoint (upper value) for each interval. Figure out the cumulative frequency and

Step-by-step explanation:

1. A city wants to reduce its carbon footprint by switching to renewable energy sources. The city's current energy consumption is measured in gigawatt hours (GWh) and they want to reduce it by 20% in the next 5 years. The city's current energy consumption is 2,000 GWh per year. What is the target energy consumption in GWh per year that the city should aim for to achieve their goal?

2. A company wants to reduce its water consumption by 50% in the next 10 years. The company currently consumes 500,000 gallons of water per day. How many gallons of water per day should the company aim to consume in 10 years to achieve their goal?

Answer:

36

Step-by-step explanation:

p/9=4

p=4*9

p=36

Answer:

Step-by-step explanation:

The length of the two diagonals of a rectangle are said to be equivalent.

From the Pythagorean theorem, the diagonal (a) will be the hypotenuse in the equation and the length (b) and width (c) will be its opposite and adjacent.

From the figure given, the length of half of two diagonal of rectangles are given.

We can simply solve for the value of x

TO = RO

3x + 1 = 2x + 4 (combine like terms by first transposing it to the other side of the equation and changing its sign from (+) to (-) or vice versa).

3x - 2x = 4 - 1

x = 3

Algebraic expression explaining real world problem is - A store is having a sale where all items are marked down 12% and an additional $2.50 is taken off the final price.

If a customer wants to find out the original price of an item before the sale, they can use the equation 12(x+2.50)=78, where x is the original price of the item.

Algebraic expressions are known as  expressions made up of :

A mathematical statement with variables, constants, coefficients, and algebraic operations is known as an algebraic expression. A good example of an algebraic expression is 5x2+6xyc. Algebraic expressions do not use equal signs, in contrast to algebraic equations. In contrast, since they are not derived from integer constants and algebraic operations, transcendental numbers like and e are not algebraic.

A store is having a sale where all items are marked down 12% and an additional $2.50 is taken off the final price. If a customer wants to find out the original price of an item before the sale, they can use the equation 12(x+2.50)=78, where x is the original price of the item.

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Answer:

----------------------------------------

Given sequence from zeros to 6th terms:

It is a GP with the common ratio of 1/2 as each term is half the previous one.

Use the nth term equation, considering the first term is t(1) = 8, and common ratio r = 1/2:

Answer:

a)  Geometric sequence

[tex]\textsf{b)} \quad t(n)=8 \left(\dfrac{1}{2}\right)^{n-1}[/tex]

Step-by-step explanation:

An Arithmetic Sequence has a constant difference between each consecutive term.

A Geometric Sequence has a constant ratio (multiplier) between each consecutive term.

From inspection of the given table, t(n) halves each time n increases by 1.

Therefore, the sequence is geometric with a common ratio of 1/2.

[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Geometric sequence}\\\\$a_n=ar^{n-1}$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\\phantom{ww}$\bullet$ $r$ is the common ratio.\\\phantom{ww}$\bullet$ $a_n$ is the $n$th term.\\\phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}[/tex]

Given:

Substitute the initial value (when n = 1) and the common ratio into the formula to create an equation for the nth term:

[tex]\implies t(n)=8 \left(\dfrac{1}{2}\right)^{n-1}[/tex]

The exact ratio of of the volume of the cone that is occupied by the sphere is 4/(1+√2)³.

What is a cone?

A cone is a three-dimensional geometric shape that narrows smoothly from a flat base (typically circular base) to a point called the apex or vertex (which creates an axis to the centre of base).

Given that the height and radius of the cone is the same.

Assume that the radius of the cone is x.

Thus the slant height of the cone is √(x² + x²) = x√2.

Assume the radius of the sphere is r.

△OEC and △BDC are similar triangles according to AAA rule.

Thus, BD/OE =BC/OC

x/r = x√2/(x -r)

Cross multiply:

x(x - r) = xr√2

x² - rx = xr√2

x² = xr√2 + rx

rx( 1+√2) = x²

r = x²/[x( 1+√2)]

r = x/ (1+√2)

The volume of the sphere is 4/3 ∏ r³ = 4/3 ∏ [ x/ (1+√2)]³

The volume of the cone is 1/3 ∏r²h =  1/3 ∏x³

The ratio of the volume of the sphere to the cone is

4/3 ∏ [ x/ (1+√2)]³  : 1/3 ∏r³

= 4 : (1+√2)³

= 4/(1+√2)³

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The derivate of function f(x)=(5x⁶+4x³)⁴is F'(x) = 4(30x³+12x²)(5x⁶+4x³)³

The derivative of a function f(x) represents its rate of change and is denoted by f'(x) or df/dx.

The derivative Formula is given as:

f1 (x)=lim△x→0 f(x+△x) − f(x)△x.

Take the derivative: F'(x) = [tex]\frac{d}{dx} (5x^6+4x^3)^5[/tex]

Substitute the inner function by (5x⁶+4x³)

Apply the chain rule:   [tex]\frac{d}{dx} (g)^4*\frac{d}{dx} (5x^6+4x^3)^4[/tex]

Calculate the derivative: [tex]4g^3*\frac{d}{dx} (5x^6+4x^3)^4[/tex]

Substitute back: [tex]4(5x^6+4x^3)^3*\frac{d}{dx} (5x^6+4x^3)[/tex]

Result after applying the chain rule: F'(x) = [tex]4(5x^6+4x^3)^3*\frac{d}{dx} (5x^6+4x^3)[/tex]

Calculate the derivative of the sum or difference:

F'(x) =[tex]4(5x^6+4x^3)^3*(\frac{d}{dx} (5x^6))+\frac{d}{dx} (4x^3))[/tex]

Take out the coefficients F'(x) = [tex]4(5x^6+4x^3)^3*(5*\frac{d}{dx} (x^6)+4\frac{d}{dx}(x^3))[/tex]

Apply the power rule: F'(x) = 4(5x⁶+4x³)³*(5*6x³+4*3x²)

Multiply the monomials: F'(x) =  4(5x⁶+4x³)³*(30x³+12x²)

Rewrite the expression: F'(x) = 4(30x³+12x²)*(5x⁶+4x³)³

So, the answer is F'(x) =4(30x³+12x²)*(5x⁶+4x³)³

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