Write and evaluate the definite integral that represents the area of the surface generated by revolving the curve on the indicated interval about the x-axis. y = 81 − x2 , −5 ≤ x ≤ 5

To find the perimeter of the regular octagon, we use the distance formula to calculate the length of one side given the endpoints and then multiply that length by eight, as there are eight equal sides in a regular octagon.

The question revolves around finding the perimeter of a regular octagon given the coordinates of one of its sides. First, we must determine the length of the side using the distance formula between the two given endpoints (-2,-4) and (4,-6). The distance formula is √((x2 - x1)^2 + (y2 - y1)^2), where (x1, y1) and (x2, y2) are the coordinates of the endpoints. After calculating the length of one side, we multiply this by 8 (since an octagon has eight equal sides) to get the perimeter.

The calculation is as follows:

The perimeter of the octagon is 16√10 units.

Answer

equilateral

Step-by-step explanation:

rate and choose brainliest

Answer:

1. 75582

2. 3047466240

Step-by-step explanation:

1.  If order does not matter, this is a combination problem.  You are choosing 8 sticks from a set of 19. (Octagons have 8 sides.)

The formula for combinations of n things chosen r at a time "n choose r" is

[tex]_nC_r = \frac{n!}{r!(n-r)!}[/tex]

[tex]_19C_8=\frac{19!}{8!(19-8)!} = \frac{19!}{8!11!}=75582[/tex]

2. If order matters, there are more possibilities. This is a permutation problem.  The number of permutations of 19 things taken 8 at a time, "19 permute 8" is

[tex]_nP_r=\frac{n!}{(n-r)!} \\ _{19}P_8=\frac{19!}{(19-8)!} = 3047466240[/tex]

Answer:

I understand the key where you represented what each of the letters represent but, there is no equation sorry I hope this helped if you would please give me a more specific equation if possible??

Answer:

2.7

Step-by-step explanation:

This can be modeled using exponential growth/decay.

A = P (1 + r)ⁿ

where A is the final amount,

P is the initial amount,

r is the rate of growth/decay,

and n is the number of cycles.

For half life problems, r = -½, and n = t / T, where t is time and T is the half life.

A = P (1 − ½)^(t/T)

A = P (½)^(t/T)

Given that P = 9, t = 10000, and T = 5730:

A = 9 (½)^(10000/5730)

A ≈ 2.7

There are approximately 2.7 mg of ¹⁴C left.

Answer: Stefan didn’t use only rigid transformations, so the figures are not congruent

Step-by-step explanation:

Stefan didn’t use only rigid transformations, so the figures are not congruent.

Two triangles are said to be congruent if their sides are equal in length, the angles are of equal measure, and they can be superimposed on each other.

In the given, Δ ABC and Δ ABD are not congruent triangles. if they are congruent then This means that the corresponding angles and corresponding sides in both the triangles are equal.

The following are the congruence theorems or the triangle congruence criteria that help to prove the congruence of triangles;

SSS (Side, Side, Side)

SAS (side, angle, side)

ASA (angle, side, angle)

AAS (angle, angle, side)

RHS (Right angle-Hypotenuse-Side or the Hypotenuse Leg theorem)

As, from the given cases the prediction of congruency of two triangles is incorrect. There is no error he made.

Hence, Stefan didn’t use only rigid transformations, so the figures are not congruent

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Answer:

Semicircle of radius of 1.6803 meters

Rectangle of dimensions 3.3606m x 1.6803m

Step-by-step explanation:

Let the radius of the semicircle on the top=r  

Let the height of the rectangle =h  

Since the semicircle is on top of the window, the width of the rectangular portion =Diameter of the Semicircle =2r

The Perimeter of the Window

=Length of the three sides on the rectangular portion + circumference of the semicircle

[tex]=h+h+2r+\pi r=2h+2r+\pi r=12[/tex]

The area of the window is what we want to maximize.

Area of the Window=Area of Rectangle+Area of Semicircle

[tex]=2hr+\frac{\pi r^2}{2}[/tex]

We are trying to Maximize A subject to [tex]2h+2r+\pi r=12[/tex]

[tex]2h+2r+\pi r=12\\h=6-r-\frac{\pi r}{2}[/tex]

The first and second derivatives are,

Area, A(r)[tex]=2r(6-r-\frac{\pi r}{2})+\frac{\pi r^2}{2}}=12r-2r^2-\frac{\pi r^2}{2}[/tex]

Taking the first and second derivatives

[tex]A'\left( r \right) = 12 - r\left( {4 + \pi } \right)\\A''\left( r \right) = - 4 - \pi[/tex]

From the two derivatives above, we see that the only critical point  of r

[tex]A'\left( r \right) = 12 - r\left( {4 + \pi } \right)=0[/tex]

[tex]r = \frac{{12}}{{4 + \pi }} = 1.6803[/tex]

Since the second derivative is a negative constant, the maximum area must occur at this point.

[tex]h=6-1.6803-\frac{\pi X1.6803}{2}=1.6803[/tex]

So, for the maximum area the semicircle on top must have a radius of 1.6803 meters and the rectangle must have the dimensions 3.3606m x 1.6803m ( Recall, The other dimension of the window = 2r)

The problem is to maximize the area of a window consisting of a rectangle and a semi-circle on top, given a fixed perimeter of framing material, which is a high school level optimization problem in geometry and calculus.

The question addresses the problem of finding the dimensions of a window with the most amount of light passing through, given a fixed amount of framing material. This is a classic problem in mathematics involving optimization under constraints, specifically related to geometry and calculus.

Let the width of the rectangle be x meters, and its height be y meters. Since the top of the window is a semi-circle, its diameter is equal to the width of the rectangle, meaning the radius of the semi-circle is x/2 meters. The perimeter of the entire window consists of the two sides and the bottom of the rectangle, and the circumference of the semi-circle. The total length of framing material is 12 meters, hence:

2y + x + (π(x/2)/2) = 12

Since the area of rectangle A = x*y and the area of the semi-circle is (x/2)²)/2, we want to maximize the total area A = x*y + (x/2)²)/2.

Using calculus, one can differentiate the area with respect to x or y and set the derivative equal to zero to find the maximum value. Assuming the student knows basic differentiation and solving equations, they can arrive at the optimal dimensions to let in the most light.

Answer:

See explaination

Step-by-step explanation:

Please kindly check attachment for the step by step solution of the given problem.

Final answer:

The hypothesis of the study is that there's a positive correlation between nose size and age, with age significantly influencing nose measurements. The strong significance level indicates a likely true relationship. This context involves genetics, evolutionary biology, and human anatomy, illustrating the complexity of evolutionary change.

Explanation:

The hypothesis being tested in the study is that there is a positive linear relationship between age and nose size. This study examines various measurements of the nose, including volume, surface area, height, and width, and assesses how these dimensions change with age. The significance level of p<0.001 suggests a very strong chance that the observed relationship is not due to random variation, but indeed reflects a true correlation between nose size and age.

Genetics and evolutionary biology play crucial roles in this context, with the Price equation outlining how certain traits might evolve over time even in the absence of natural selection. Importantly, reification can lead to misconceptions as it implies that there is a direct correspondence between a named entity, like 'nose genes', and physical traits.

An understanding of human anatomy, like the relationship between the nasal aperture and the overall size of the nose—exampled by the large noses of Neanderthals, reflects broader principles of evolutionary change and adaptation. This complex interplay of genetics, physical development, and evolutionary factors highlights the importance of incremental transformations and selection in shaping human features over time.

Answer:

1/60 probabiliity

Step-by-step explanation:

You have two independent events that you want to put together.

Let Pr. mean "probability"

Pr(5 from 10 cards and 2 on a dice ) = Pr(5 from 10 cards) * Pr( 2 on a dice)

Pr(5 from 10 cards and 2 on a dice ) =(1/10) * (1/6)

= 1/60

Pr(5 from 10 cards and 2 on a dice ) =0.0167

or 1.67% probability

Answer:

x = domain

y = range

Step-by-step explanation:

Given:

Given that Steven purchased a box of chocolate shaped like a square pyramid.

The box is 12 inches tall and the area of the bottom of the box is 35 square inches.

We need to determine the expression that is used to find the number of chocolates that the box holds.

Expression:

The expression that is used to find the number of chocolates that the box holds can be determined using the formula,

[tex]V=\frac{1}{3} Bh[/tex]

where B is the area of the base and h is the height of the pyramid.

Substituting B = 35 and h = 12 in the above formula, we get;

[tex]V=\frac{1}{3}(35 \cdot 12)[/tex]

Thus, the expression that is used to find the number of chocolates that the box holds is [tex]\frac{1}{3}(35 \cdot 12)[/tex]

Hence, Option b is the correct answer.

Answer:

Option 2

Step-by-step explanation:

Volume of a pyramid

⅓ × base area × height

⅓ × 35 × 12

12 × 35 × ⅓

Answer:

Step-by-step explanation:

The school play ground is.

5ft long rectangle

The designer want to increase the length,

Then, there will be 3 identical rectangle of total length of 17ft

By how much did he increase the of each rectangle

Original, the length of the rectangle is 5ft, now, we want to design three rectangle. This three identical rectangle will have total length of (5+5+5)ft,

The original length of the three rectangle is meant to be 15ft,

Now he increase the length to 17ft,

This shows that he has increase all the three rectangle by 2ft.

So, each rectangle is expected to be increased by ⅔ft

So, he increased each rectangle length by ⅔ft.

Mathematically,

Let the increase be x

Then, he increase each length by x, so the new length is

5+x

Then, there are 3 rectangles, so their total length after increase is 3×(5+x)

And this total length after increase is give as 17ft

Then,

3(5+x) = 17

Solving this

15 + 3x = 17

3x = 17-15

3x = 2

x = ⅔ ft

So, the mathematical model of the increase is 3(5+x) = 17 and the increase is ⅔ft

Answer:

Communitive property of addition

Step-by-step explanation:

Answer:

The predicted percentage vote for Al Gore for the county with 44.0954% African Americans is 25.76%.

Step-by-step explanation:

The least square regression line is used to determine the relationship between a response variable (or dependent variable) and an explanatory variable (or independent variable).

The least square regression line can be used to predict the future value or estimate the past value of the dependent variable based on the independent variable.

The general form of a least square regression line is:

[tex]y=\alpha +\beta x[/tex]

Here,

y = response variable

x = explanatory variable

α = intercept

β = slope

The least square regression line for predicting the percentage vote for AI Gore (y) from the percentage of county residents who were African American (x) is given by:

[tex]\hat y=25.4495+0.6956 x[/tex]

Compute the predicted value of y for x = 44.0954% = 0.440954 as follows:

[tex]\hat y=25.4495+0.6956 x[/tex]

  [tex]=25.4495+0.6956 \times 0.440954\\=25.4495+0.3067276024\\=25.7562276024\\\approx 25.76[/tex]

Thus, the predicted percentage vote for Al Gore for the county with 44.0954% African Americans is 25.76%.

Using the equation of the estimated regression line, the predicted percentage vote for Al Gore in Baker County, Georgia is approximately 56.3555% based on a population that was 44.0954% African American.

In this scenario, we have a linear regression line of the form y^ = 25.4495 + 0.6956x, where y^ represents the estimated percentage vote for Al Gore, and x represents the percentage of county residents who are African American. We are given that in Baker County, Georgia, 44.0954% of residents are African American. By substituting this x value into the equation, we can find the predicted y value:

y^ = 25.4495 + 0.6956*(44.0954)

This gives an estimated vote for Al Gore in Baker County as y^ = 56.3555 approximately, to four decimal places. This implies that the model predicts that about 56.3555% of the vote in Baker County went to Al Gore.

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The total distance covered is 6 m

Step-by-step explanation:

Let the total distance be '2d'

Total time = 1.25hrs

Downhill speed = 6 mphr

Uphill speed = 4 mphr

(d/6) +(d/4) = 1.25

(2d + 3d) /12 = 1.25

5d/12 = 1.25

d = 3m

So, 2d = 6 m

The total distance covered is 6 m

Answer:

6

Step-by-step explanation:

A triangular prism, a shape with two identical triangular faces and three identical rectangular faces, has 6 vertices where the edges of these faces meet.

In geometry, a shape's vertices are the points where two or more lines or edges meet. When we consider a triangular prism, we see that it is a figure with two triangular faces and three rectangular faces. Each triangular face has 3 vertices, and since these faces are identical, they share vertices. If you count up all these points, we find that a triangular prism has 6 vertices in total.

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Answer:

6.3972

Step-by-step explanation:

This is a normal distribution problem.

-We claim that most adults are more likely to erase their data;

[tex]p_o>0.5\\\\H_o:p>0.5[/tex]

-The test statistic for a stated hypothesis  for proportions is given by the formula:

[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o(1-p_o}{n}}}[/tex]

Given the size of the random sample is 522 and that 62% of them are susceptible to erasing their data.

-Let [tex]\hat p[/tex] be the sample proportion. The value of the test statistic :

[tex]z=\frac{0.64-0.5}{\sqrt{\frac{0.5\times 0.5}{522}}}\\\\=6.3972[/tex]

Hence, the test statistic is z=6.3972

In Ebru’s deck of cards, the probabilities of choosing a 2 and choosing a spade are respectively 1/13 and 1/4. However, these events are not independent, so the probabilities impact each other. The statements (a), (b), (c), and (e) are not true, while (d) is true.

The deck of 52 cards contains 4 suits and each suit has 13 cards, meaning there are 4 cards of 2 and 13 spades in total. We can use these numbers to calculate the probabilities.

When it comes to P(A | B)=P(A) and P(B | A)=P(B), these would hold true if A and B were independent events, meaning the occurrence of one does not affect the probability of the other happening. However, these events are not independent. If Ebru picks a 2, the chances of her picking a spade change, and similarly whether she picks a spade affects the chances of her picking a 2.

So that means (a) and (b) are not true, while (c) is not true because events A and B do indeed influence each other, making (d) true. If A and B were independent, then (e) would be true as the probability of both events A and B happening would be multiplied. However, in this case, they influence each other so (e) is not true.

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From this analysis, the true statements are a, b, c, and e. The decision of a card being a 2 and the choice of a card being a spade are independent events. The outcomes are not dependent on each other.

From a standard deck of cards, we have 4 cards that are 2s (one in each of the four suits: hearts, clubs, diamonds, and spades) and 13 cards that are spades (including numbered 2-10, a jack, a queen, a king, and an ace). Therefore, the probability P(A) that Ebru selects a 2 is 4 out of 52 (or 1/13), and the probability P(B) that she selects a spade is 13 out of 52 (or 1/4).

a) P(A | B) refers to the probability that Ebru selects a 2 given that she has already chosen a spade. In this case, there is only one 2 among the 13 spades, so P(A | B) = 1/13 which is equal to P(A). Hence, statement a is true.

b) P(B | A) is the probability that Ebru selects a spade having already chosen a 2. There is one spade among the 4 twos, so P(B | A) = 1/4, which is equal to P(B).  Hence, statement b is true.

c) We can see that both P(A | B) = P(A) and P(B | A) = P(B). Therefore, events A and B are independent, making statement c true.

d) Because A and B are independent, statement d, which suggests that they are dependent, is false.

e) For independent events, P(A and B) = P(A) * P(B), which is (1/13) * (1/4), so statement e is true.

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Answer:

z1 = 7.71 + 0.02 i

z2 = 7.73 + 0.306 i

z3 = 7.78 + 0.59 i

Step-by-step explanation:

To find the roots you use:

[tex]z^{\frac{1}{n}}=r^{\frac{1}{n}}[cos(\frac{\theta+2\pi k}{n})+isin(\frac{\theta+2\pi k}{n})][/tex]     ( 1 )

n: the order of the roots

k: 0,1,2,...,n-1

First, you write z in polar notation:

[tex]z=re^{i\theta}\\\\r=\sqrt{(473)^2+(4)^2}=473.01\\\\\theta=tan^{-1}(\frac{4}{473})=0.48\°[/tex]

Thus, by using these values for the angle and r in the expression (1), you obtain:

[tex]k=0\\\\z_1=(473.01)^{1/3}[cos(\frac{0.48+2\pi(0)}{3})+isin(\frac{0.48+2\pi(0)}{3})]\\\\z_1=7.79(0.99+i2.79*10^{-3})=7.71+i0.02\\\\z_2=7.79[cos(\frac{0.48+2\pi(1)}{3})+isin(\frac{0.48+2\pi(1)}{3})]\\\\z_2=7.73+i0.306\\\\z_3=7.79[cos(\frac{0.48+2\pi(2)}{3})+isin(\frac{0.48+2\pi(2)}{3})]\\\\z_3=7.78+i0.59[/tex]

hence, from the previous results you obtain:

z1 = 7.71 + 0.02 i

z2 = 7.73 + 0.306 i

z3 = 7.78 + 0.59 i

I attached and image of the plot

Answer:

4 hours

Step-by-step explanation:

We know that there are 24 hours in a day. Therefore, we will subtract from 24.

If mark spends one-third of his day sleeping, then he will be asleep for 8 hours.

[tex]\frac{1}{3}\times24 \\=\frac{24}{3} \\=8[/tex]

We also know that if Mark spends one-sixth of his day at soccer practice, he will have been practicing for 4 hours.

[tex]\frac{1}{6}\times24 \\=\frac{24}{6} \\=4[/tex]

We now simply subtract from 24:

[tex]\text{hours in a day - hours asleep - hours at school - hours practicing} \\24 - 8 - 8 - 4 \\= 4[/tex]

Therefore, Mark will have 4 hours of free time.

Answer:

133/1152

Step-by-step explanation:

A probability is the number of desired outcomes divided by the number of total outcomes.

In this question:

19+15+14 = 48 marbles

Of which 19 are red, 15 are yellow and 14 are blue.

What is the probability of pulling out a blue marble, followed by a red marble, if you replace the blue marble first

Blue marble

48 marbles, of which 14 are blue.

So [tex]P_{A} = \frac{14}{48} = \frac{7}{24}[/tex]

Red marble

48 marbles, of which 19 are red.

So [tex]P_{B} = \frac{19}{48}[/tex]

Both:

[tex]P = P_{A} \times P_{B} = \frac{7}{24} \times \frac{19}{48} = \frac{133}{1152}[/tex]

So the correct answer is:

133/1152

Answer:

r = 4.5m

d = 57 yds

Step-by-step explanation:

The radius is 1/2 of the diameter

d= 9 meter

  r = d/2  =9/2 = 4.5 m

The diameter is twice the radius

r = 28.5 yds

d = 2r = 2*28.5 =57 yds

Answer:

a) r=4.5 m

b) d= 57 yd

Step-by-step explanation:

a)

The diameter is twice the radius, which can be written as:

d=2r

We know the diameter is 9, so we can substitute that in for d.

9=2r

To solve for r, we need to isolate the variable. To do this, divide both sides by 2

9/2=2r/2

4.5=r

So, the radius is 4.5 meters

b)

The diameter is twice the radius, or

d=2r

We know the radius is 28.5, so we can substitute that in for r

d=2*28.5

Multiply

d=57

So, the diameter is 57 yards

Answer:

C. 40%

Step-by-step explanation:

Using set notations,

Check the attachment for the diagram.

Let the total fund shared by the town be 100% which will be our universal set.

Let X be proportion of households that own mutual funds but not individual

From the venn diagram,  

The total number of people that owned mutual fund M = (proportion of  households that owned both mutual fund and individual stock) +  (proportion of households that own mutual funds but not individual stocks)

If X is the proportion of households that own mutual funds but not individual stocks

The total number of people that owned mutual fund = (proportion of  households that owned both mutual fund and individual stock) +  X

X = (the total number of people that owned mutual fund)- (proportion of  households that owned both mutual fund and individual stock)

X = 60% - 20%

X = 40%

Final answer:

To calculate the proportion of households owning only mutual funds, subtract the percentage owning both mutual funds and individual stocks from the percentage owning mutual funds, giving us 40%. The answer is C) 40%.

Explanation:

The question provided falls under the category of probability and sets in mathematics. It involves understanding how to calculate the proportion of households that own mutual funds but not individual stocks when certain percentages are provided for mutual fund ownership, stock ownership, and those owning both. This is commonly known as a problem involving the use of Venn Diagrams or set theory.

Firstly, it is stated that 60% of households own mutual funds and 40% own individual stocks. Among them, 20% own both mutual funds and individual stocks. To find the proportion of households that own only mutual funds (and not individual stocks), we subtract the percentage that owns both from the percentage that owns mutual funds. Therefore:

Proportion owning only mutual funds = (Percentage owning mutual funds) - (Percentage owning both mutual funds and stocks)

This gives us:

Proportion owning only mutual funds = 60% - 20% = 40%

The correct answer is C) 40%.

Answer:

Prism A

Step-by-step explanation:

Volume of a triangular prism is calculated like this : 1/2 × 7 × 4 × 9 = 126 m^3

Volume of a cuboid is calculated like this : 8 × 8 × 2 = 128 m^3

Answer:

A

Step-by-step explanation:

Answer:

11

Step-by-step explanation:

8x - 3(-z - y)

Substitute

8(1) - 3(-(-2) - 3)

Simplify

8 - 3(-1)

Multiply

8 + 3

Add

11

Answer:

Here; 8x-3(-z-y) =

so your asking 8x-3(-z-y) right?

Let's simplify step-by-step.

8x−3(−z−y)

awnser again xD

=8x+3y+3z

Final answer:

The approximate probability of winning 4 games in a row when each game has a 1/3 chance of winning is 1.23%.

(Option B)

Explanation:

To find the approximate probability of winning 4 games in a row, we need to multiply the probabilities of winning each game. Since the probability of winning each game is 1/3, we can calculate the overall probability as [tex](1/3)^4[/tex]. Using a calculator, this comes out to be approximately 0.0123 or 1.23%.

Multiplying the individual probabilities of winning each game, given as 1/3, results in the overall probability of winning 4 games in a row, expressed as [tex](1/3)^4[/tex]. Using a calculator, this evaluates to approximately 0.0123, or 1.23%, highlighting the cumulative nature of independent events.

Answer:

The value of t test statistics is -2.

Step-by-step explanation:

We are given that a study was conducted to determine whether UH students sleep fewer than 8 hours.

The study was based on a sample of 100 students. The sample mean number of hours of sleep was 7 hours and the sample standard deviation was 5 hours.

Let [tex]\mu[/tex] = mean number of hours UH students sleep.

So, Null Hypothesis, [tex]H_0[/tex] : [tex]\mu \geq[/tex] 8 hours     {means that UH students sleep more than or equal to 8 hours}

Alternate Hypothesis, [tex]H_A[/tex] : [tex]\mu[/tex] < 8 hours     {means that UH students sleep fewer than 8 hours}

The test statistics that would be used here One-sample t test statistics as we don't know about the population standard deviation;

                         T.S. =  [tex]\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }[/tex]  ~ [tex]t_n_-_1[/tex]

where, [tex]\bar X[/tex] = sample mean number of hours of sleep = 7 hours

             s = sample standard deviation = 5 hours

            n = sample of students = 100

So, test statistics  =  [tex]\frac{7-8}{\frac{5}{\sqrt{100} } }[/tex]  ~ [tex]t_9_9[/tex]

                              =  -2

Hence, the value of t test statistics is -2.

Answer:

=3(x^2 − 7)

Step by Step explanation:

3x^2 − 21

=3(x^2 − 7)

Answer:

=3(x^2 − 7)

Step-by-step explanation:

Answer: The answer would be D

Step-by-step explanation:

This is because all of the models on the graph show in the same place having the same middle but at different heights but not the same standard deviation.

Each distribution has the same mean and a different standard deviation is the correct statement about the graph.

Three different normal distributions are given in the graph.

The spread in the given normal curves is due to different standard deviation.

The mean for all the three curves lies at the center of the curve.

Therefore, the graph shows that each distribution has the same mean and a different standard deviation.

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The complete question is given below.

The graph below shows three different normal distributions.

Which statement must be true?

Each distribution has a different mean and the same standard deviation.

Each distribution has a different mean and a different standard deviation.

Each distribution has the same mean and the same standard deviation.

Each distribution has the same mean and a different standard deviation.