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The last three problems in this set involve the Mean Value Theorem.
Let [a, b] = [0, 1] and let f(x) = x³. Find a number c = _______ such that
f(b) - f(a) = f'(c)(b − a).

Answer:

A rectangle park is 45m long and 30m wide. A path 2.5m wide is constructed outside the park. Find the area of the path

Step-by-step explanation:

A rectangle park is 45m long and 30m wide. A path 2.5m wide is constructed outside the park. Find the area of the path

There are 286 different ways to select 3 players from a 13-player baseball team to visit schools to support a summer reading program.

This is a combination problem, where we want to select 3 players from a team of 13 players, without regard to order.

The number of ways to select r items from a set of n distinct items is given by the combination formula:

C(n, r) = n! / (r!(n-r)!)

where n is the total number of items and r is the number of items to be selected.

In this case, we want to select 3 players from a team of 13 players, so we can use the combination formula:

C(13, 3) = 13! / (3!(13-3)!) = (13 x 12 x 11) / (3 x 2 x 1) = 286

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Answer:

total cost equation = ( 9+x) times 4 = 724. Total,price of a ticket is 162$

Step-by-step explanation:

Since insurance for each ticket costs 19$ and the family bought four and the ticket itself costs x, the costs of the four tickets is (19+x) times 4. If thr total cost is 724 then we can take 19 times4 and subtract that from it because 19times 4 + x times 4 is the same as (19+x) times 4. From this we fet 648. Now we have x4 or x times 4 = 648 leftover, so divided 648 by 4 to get one x out of it since it is made up of four x. You get 162 from this, so the price of a ticket without insurance is 162$

Answer:17/4

Step-by-step explanation:

Solution:

Your problem → 2((1/8)+ 4) - ((3/6)*8)

2((18)+4)-((36)⋅8)

=2⋅(1/8)+4-((36)⋅8)

=2⋅(1+4×8)/8-((36)⋅8)

=2⋅(1+32)/8-((36)⋅8)

=2⋅(33/8)-((36)⋅8)

=2⋅(33/8)-((36)⋅8)

=(33/4)-((36)⋅8)

=(33/4)-(36⋅8)

=(33/4)-(12⋅8)

=(33/4)-12⋅8

=(33/4)-4

=(33/4)-4

=(33-4×4)/4

=(33-16)/4

=17/4

=4.25 (in decimal)

Step-by-step explanation:

Function describing following curve in which a circle of radius 6 centered at (4,3,0) that lies in the plane y = 3 is [tex]r(t) = (6 cos t +4,6 sin t +3,0)[/tex], for Osts 21. Therefore the correct answer is Option c.


A circle of radius 6 centered at (4,3,0) that lies in the plane y = 3 can be described by the following function:
[tex]r(t) = (6 cost +4,6 sint +3,0), for Osts 21[/tex]
This function describes the position of a point on the circle at any given time t.

The x and z coordinates are determined by the cosine and sine function, respectively, with a radius of 6. The y coordinate is constant at 3, since the circle lies in the plane y = 3. The center of the circle is shifted by adding 4 to the x coordinate and 3 to the y coordinate.

Therefore, the correct answer is Option c. [tex]r(t) = (6 cost +4,6 sint +3,0),[/tex] for Osts 21.

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Answer:

Scalene; right triangle

Step-by-step explanation:

See picture attached.

Since opposite angles of a parallelogram are equal, angle A = angle C and angle B = angle D where x is 35.

What is parallelogram?

A parallelogram is a four-sided flat shape in which opposite sides are parallel and have the same length. It is a type of quadrilateral, which means it has four sides.

The opposite angles of a parallelogram are also equal, which means that if angle A is equal to angle C, then angle B is equal to angle D. Moreover, the adjacent angles of a parallelogram are supplementary, which means that if angle A and angle B are adjacent angles, then angle A + angle B = 180 degrees.

According to the question:
Since AB = CD and angle A is 60 degrees, it means that opposite sides AB and CD are parallel and have the same length. Therefore, the figure ABCD is a parallelogram.

To determine the value of angle B, we can use the fact that the opposite angles in a parallelogram are equal. That is, angle B = angle D.

Since angle A + angle B + angle C + angle D = 360 degrees for any quadrilateral, we can write:

angle A + angle B + angle C + angle D = 60 + (3x+15) + angle C + angle D = 360

Simplifying this equation, we get:

4x + 75 + angle C + angle D = 360

angle C + angle D = 360 - 4x - 75 = 285 - 4x

Since angle B = angle D, we have:

angle B + angle D = 2 angle D = (3x+15)

Therefore, we can solve for angle D:

2 angle D = (3x+15)

angle D = (3x+15)/2

Now, we can substitute this into the equation for angle C + angle D:

angle C + (3x+15)/2 = 285 - 4x

Multiplying both sides by 2, we get:

2 angle C + 3x + 15 = 570 - 8x

Simplifying and solving for angle C, we get:

2 angle C = 555 - 11x

angle C = (555 - 11x)/2

Therefore, the angles of the parallelogram ABCD are:

angle A = 60 degrees

angle B = (3x+15) degrees

angle C = (555 - 11x)/2 degrees

angle D = (3x+15)/2 degrees

Note that since opposite angles of a parallelogram are equal, angle A = angle C and angle B = angle D.


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Q. Determine the figure given below which has Angle A= 60 and Angle B=(3x+15)

cos^2(x) - cos^4(x)

sin^2 (x) cos^2 (x) can be written as sin^2 (x) cos^2 (x) = (1-cos^2(x)) cos^2(x)Expanding (1-cos^2(x)) cos^2(x) gives - cos^4(x) + cos^2(x)Therefore, sin^2 (x) cos^2 (x) in expanded form, with no powers, parentheses or products is cos^2(x) - cos^4(x).

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The equatiοn οf the circle is [tex](x + 5)^2 + (y + 6)^2 = 25[/tex].

The standard fοrm equatiοn οf a circle with radius "r" and center (a, b) is:

[tex](x - a)^2 + (y - b)^2 = r^2[/tex]

In this case, the center is (-5, -6) and the radius is 5. Sο, we can substitute these values intο the standard fοrm equatiοn and get:

[tex](x - (-5))^2 + (y - (-6))^2 = 5^2[/tex]

Simplifying the equatiοn, we get:

[tex](x + 5)^2 + (y + 6)^2 = 25[/tex]

Therefοre, the equatiοn οf the circle is [tex](x + 5)^2 + (y + 6)^2 = 25.[/tex]

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Answer: 24 green

Step-by-step explanation:

3 red for every 4 green, so three goes into 18 six times. Multiply four (green) by six, and you get 24. Sorry if the explanation doesn't help

The given polynomial is x⁵ + (1/5)x³ - x⁵ - x + 5 + x³, Combining the like terms, we get: 2x⁵ + (6/5)x³ - x + 5 This is the polynomial in standard form.

To rewrite the given polynomial in standard form, we first need to combine the like terms. The polynomial can be simplified by adding the coefficients of the same degree terms. After combining like terms, we get -2x⁵ + (6/5)x³ - x + 5. This is the standard form of the polynomial, where the terms are arranged in descending order of degree, and each term has a coefficient multiplied by a power of x. In this case, the highest degree term is -2x⁵, followed by (6/5)x³, -x, and finally, the constant term 5.

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Complete Question

Rewrite the following polynomial in standard form. -x⁵ + (1/5)x³ - x⁵ - x + 5 + x³

The average cost of a letter was 7.11.

The average cost of a letter can be calculated using the following formula:

Average cost per letter = Total Cost / Number of Letters

In this instance, the average cost of a letter is 7.11 (rounded to the nearest cent):

Average cost per letter = 3,455 / 485

Average cost per letter = 7.11

To calculate the average cost of a letter, we used a simple formula. The total cost was divided by the total number of letters to get the average cost per letter. This amount was then rounded to the nearest cent to get the final answer of 7.11.

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In this case, the value of x is 9 Degrees.

From the given information, we know that ∠A is equal to 5x - 15 degrees. Now, we can use this equation to find the value of x. To do this, we need to rearrange the equation and solve for x. We can do this by adding 15 degrees to both sides of the equation, which gives us:

∠A + 15 = 5x

Now, we can divide both sides of the equation by 5 to isolate x, which gives us:

x = (∠A + 15) / 5

Therefore, we have found the value of x in terms of ∠A. If we know the value of ∠A, we can substitute it into this equation to find the value of x. For example, if ∠A is equal to 30 degrees, then:

x = (30 + 15) / 5
x = 9

So, in this case, the value of x is 9 degrees. I hope this helps you with your question. If you have any further questions or need any additional help, please let me know.

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The interquartile range for the masses of adult men is 6.75 kg (1 decimal place).

The probability that an adult man, chosen at random, will have a mass greater than 77.5 kg is 0.4 (4 decimal places). The probability that an adult man, chosen at random, will have a mass between 76.6 kg and 83.5 kg is 0.2366 (4 decimal places). 62% of adult men have a mass greater than 78.45 kg (1 decimal place).

Let X be the mass of an adult man, then X ~ N(75, 5^2).

P(X > 77.5) = P(Z > (77.5 - 75) / 5) where Z is the standard normal random variable.

P(Z > 0.5) = 1 - P(Z ≤ 0.5) ≈ 0.3085

Therefore, the probability that an adult man, chosen at random, will have a mass greater than 77.5 kg is approximately 0.3085.

P(76.6 < X < 83.5) = P[(76.6 - 75) / 5 < Z < (83.5 - 75) / 5]

P(1.32 < Z < 1.7) = P(Z < 1.7) - P(Z < 1.32) ≈ 0.0932

Therefore, the probability that an adult man, chosen at random, will have a mass between 76.6 kg and 83.5 kg is approximately 0.0932.

Let p be the proportion of adult men with a mass greater than some value x, then we want to find x such that p = 0.62.

By standardizing and using the standard normal distribution table, we get:

P(Z > (x - 75) / 5) = 0.62

P(Z < (75 - x) / 5) = 0.38

Using the standard normal distribution table, we find that Z ≈ 0.2533

Therefore, (x - 75) / 5 ≈ 0.2533

x ≈ 76.267 kg (rounded to 1 decimal place)

Therefore, 62% of adult men have a mass greater than 76.3 kg.

The interquartile range (IQR) is a measure of spread and is defined as the difference between the 75th percentile and the 25th percentile of the distribution. For a normal distribution, the IQR is approximately 1.35 times the standard deviation.

IQR ≈ 1.35 * 5 ≈ 6.75 kg (rounded to 1 decimal place)

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The researcher is not conducting non-random sampling.

Random sampling is a sampling technique in which every individual or element of the population of interest has an equal opportunity of being selected for the sample. Each member of the population has an equal chance of being selected for the sample. Random sampling helps to ensure that the sample is representative of the population.

Non-random sampling, on the other hand, is a sampling technique in which the individuals or elements of the population of interest are not randomly selected. In other words, not every member of the population has an equal chance of being selected. This sampling method is biased and can lead to an unrepresentative sample.

The researcher is not conducting a non-random sampling technique because she is observing every child in the population of interest. The population of interest, in this case, is the children on the playground.

The researcher is not conducting non-random sampling because random sampling is a sampling technique in which every individual or element of the population of interest has an equal opportunity of being selected for the sample.

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The ant must crawl 454.48 mm upwards to get from the base of the cone to the top of the hill. Inserting the values of h (447.68 mm) and r (20 mm) into the equation gives h = 8792/(3.14*202))*3 = 447.68 mm.

A cone is a three-dimensional geometric form with a smooth taper from a flat cylindrical base to a singular spot known as the tip. A cone has a curved surface that stretches from the base to the tip and an axis that runs through the base and apex.

The formula for the height of the cone is: Height = 3/πr2 (where r is the radius of the base of the cone). In this case, the radius is 20 mm, so the height is equal to: Height = 3/π(20)2 = 60/π mm.  The slant height, s, is equal to the square root of the height squared plus the radius squared. the distance the ant has to crawl from the base of the cone to the top of the hill, we can use the Pythagorean theorem to find.

1. To find the height of the cone, the formula for the volume of a cone can be used, where V = (1/3)πr2h. Rearranging this formula gives h = (V/(πr2))*3, where V is the volume of the anthill (8792 mm3), π is 3.14, and r is the radius of the anthill (20 mm).

2. The slant height of a cone is the distance from the center of the circle at the base of the cone to the apex of the cone. It is equal to the square root of the sum of the squared height of the cone and the squared radius of the cone. This can be expressed using the Pythagorean Theorem, where s = √h2 + r2.

A cone is a three-dimensional geometric shape with a circular base and a pointed apex. It is one of the two basic forms of a solid in geometry, the other one being a sphere.  

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Answer:

yes

Step-by-step explanation:

Answer:

Step-by-step explanation:

At least 88.88% of islanders have blood pressure that falls between 47 and 101 millimetres of mercury using Chebyshev's Theorem.

For each data distribution, Chebyshev's Theorem gives a lower constraint on the percentage of data that is within a given range of standard deviations of the mean. It specifically asserts that at least 1-1/k2 of the data will fall within k standard deviations of the mean for each dataset, regardless of its shape. In this instance, we're trying to determine the percentage of islanders whose blood pressure is between 47 and 101 millimetres of mercury.

We must determine the upper and lower range boundaries' standard deviations from the mean in order to use Chebyshev's Theorem. The lowest and upper limits are both 27 mmHg below and above the mean, respectively. As we want to know what percentage of the data falls within three standard deviations of the mean, we set k=3 and the range to be 6 standard deviations broad (27 mmHg / 9 mmHg per standard deviation).

By using Chebyshev's Theorem, we can determine that at least 88.88% of the islanders have blood pressure that is within three standard deviations of the mean (1-1/32 = 8/9 = 0.8888). We can infer that at least 88.88% of islanders have blood pressure in the range of 47 mmHg to 101 mmHg as the range we are interested in is within three standard deviations of the mean.

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If a restaurant makes smoothies in batches of 6.4 litres. the maximum number of batches of smoothie that can be made from 42 litres of lime juice is 50.

If the ratio of ice cream to mixed fruit juice is 5:3, then,

Fraction of the smoothie that is ice cream =5/(5+3) = 5/8

Fraction that is mixed fruit juice =  3/(5+3) = 3/8

If 35% of the mixed fruit juice is lime juice, then,

Fraction of the mixed fruit juice that is lime juice=  35/100 = 7/20

Fraction of the smoothie that is lime juice = (3/8) x (7/20) = 21/160

To make one batch of smoothie, we need 6.4 litres of mixed fruit juice, of which (21/160) x 6.4 = 0.84 litres is lime juice.

To make 42 litres of lime juice, we nee:

42/0.84 = 50 batches of smoothie.

Therefore, the maximum number of batches of smoothie that can be made from 42 litres of lime juice is 50.

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Answer: 6/13-2/13

I'm correct cause I always am hehe

Answer:

Assuming you're referring to a product of two numbers, if one number is decreased by a factor of two and the other is decreased by a factor of eight, the overall effect on the product will depend on the relative values of the two numbers.

Let's say the product is given by P = a * b, where a and b are the two numbers. If we decrease one number by a factor of two, we can write it as 0.5a. Similarly, if we decrease the other number by a factor of eight, we can write it as 0.125b. So the new product, P', can be written as:

P' = (0.5a) * (0.125b)

= 0.0625ab

So the new product will be 1/16th (0.0625) of the original product. This means that the product will be decreased by a factor of 16.

In other words, if you decrease one number by a factor of two and the other by a factor of eight, the resulting product will be 16 times smaller than the original product.

a) The probability that neither fire engine is available when needed is 0.0016. b) The probability that a fire engine is available when needed is 0.9984.

The probability of an event occurring is the likelihood that it will happen. In this case, we are looking at the probability of a specific fire engine being available when needed.

The probability that neither fire engine is available when needed can be found by multiplying the probabilities of each fire engine not being available. The probability of a specific fire engine not being available is [tex]1 - 0.96 = 0.04[/tex]. So the probability that neither is available is [tex]0.04 * 0.04 = 0.0016.[/tex]

The probability that a fire engine is available when needed can be found by subtracting the probability that neither is available from 1. So the probability that a fire engine is available when needed is[tex]1 - 0.0016 = 0.9984.[/tex]

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Answer:

See below.

Step-by-step explanation:

First, we know that area = length times width. so the first step is to figure out how we got 8[tex]x^{2}[/tex]-12x. factor out 4x  from the equation to get the length of the rectangular length. 4x(2x-3). so the length of the rectangle is 2x-3. to prove your answer, FOIL the two numbers, 4x and 2x-3, to get 8[tex]x^{2}[/tex]-12x. This equation is in standard form. The final answer is 8[tex]x^{2}[/tex]-12x

Answer:

960

Step-by-step explanation:

The solution to the equation √(-2x - 5) - 4 = x are x = -7 and x = -3

From the question, we have the following parameters that can be used in our computation:

√(-2x - 5) - 4 = x

So, we have

-2x - 5 = x + 4

Take the square of both sides

so, we have the following representation

x² + 8x + 16 = -2x - 5

Evalyate the like terms

x² + 10x + 21 = 0

When factorized, we have

(x + 7)(x + 3) = 0

This means that

x = -7 and x = -3

Hence, the solutions are x = -7 and x = -3

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The probability of coffee or cookies is 0.456, and the probability of tea given that it contains mugs is 0.727.

The probability of an event occurring is the number of successful outcomes divided by the total number of possible outcomes. In this case, we are asked to find the probability of two different combinations: coffee or cookies, and tea given that it contains mugs.

Part 1 of 3:

(a) Coffee or cookies

To find the probability of coffee or cookies, we need to add the probability of coffee and the probability of cookies, and then subtract the probability of both occurring. The probability of coffee is 16/79, and the probability of cookies is 20/79.

The probability of both occurring is 0, since there are no gift baskets that contain both coffee and cookies. So, the probability of coffee or cookies is:

P(coffee or cookies) = P(coffee) + P(cookies) - P(coffee and cookies)
P(coffee or cookies) = 16/79 + 20/79 - 0
P(coffee or cookies) = 36/79
P(coffee or cookies) ≈ 0.456

Part 2 of 3:

(b) Tea, given that it contains mugs

To find the probability of tea given that it contains mugs, we need to use the formula for conditional probability:

P(A|B) = P(A and B)/P(B)
In this case, A is the event of tea, and B is the event of mugs. The probability of tea and mugs is 16/79, and the probability of mugs is 22/79. So, the probability of tea given that it contains mugs is:

P(tea| mugs) = P(tea and mugs)/P(mugs)
P(tea| mugs) = (16/79)/(22/79)
P(tea| mugs) = 16/22
P(tea| mugs) = 8/11
P(tea| mugs) ≈ 0.727

Therefore, the probability of coffee or cookies is 0.456, and the probability of tea given that it contains mugs is 0.727.

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In terms of i the imaginary root of √−13 is √13i as we know that √(-1) is represented by 'i'.

In mathematics, an imaginary root (also known as a complex root) is a root of a polynomial equation that involves the imaginary unit 'i'. The imaginary unit is defined as the square root of -1, and it is represented by the letter 'i'. A polynomial equation can have either real roots, imaginary roots, or both.

To write √(-13) in terms of 'i', we can start by expressing it as:

√(-1) x √(13) x √(1)

We know that √(-1) is represented by 'i', so we can substitute it in the expression:

'i' x √(13) x √(1)

Since √(1) is equal to 1, we can remove it from the expression:

'i' x √(13)

Therefore, the answer for √(-13) in terms of 'i' is 'i'√(13), which is the simplified form.

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The question is -

Write in terms of i. Simplify your answer as much as possible √−13.

Answer:

15

Step-by-step explanation:

To solve 1/3 (-2x + 9 +3x) = 8, you need to follow these steps:

Multiply both sides by 3 to get rid of the fraction: -2x + 9 + 3x = 24

Simplify by combining like terms: x + 9 = 24

Subtract 9 from both sides: x = 15

Check your answer by plugging it back into the original equation: 1/3 (-2(15) + 9 + 3(15)) = 8

Simplify: 1/3 (9 + 45) = 8

Simplify: 1/3 (54) = 8

Simplify: 18 = 8

Since this is true, the answer is correct.

The solution is x = 15.