Find the area of the shaded regions:

Answer:

[tex]f(g(x))=x^{6}-19x^{3}+90[/tex]

Step-by-step explanation:

We have been given the following equations;

f(x)=x^2−3x+2

g(x)=x^3−8

We are required to determine the composite function (f⋅g)(x). This simply means that we shall substitute the function g(x) in place of x in the function f(x);

[tex]f(g(x)) = g(x)^{2}-3g(x)+2\\\\f(g(x))=(x^{3}-8)^{2}-3(x^{3}-8)+2\\\\f(g(x))=x^{6}-16x^{3}+64-3x^{3}+24+2\\\\f(g(x))=x^{6}-19x^{3}+90[/tex]

The correct statement is: Angle G is congruent to angle P.

Congruency in geometry refers to the similarity between two shapes or figures. When two shapes are congruent, their corresponding angles and sides are equal in measure, preserving the same shape and size but allowing for possible reflection or rotation.

The correct statement that compares the angles is: Angle G is congruent to angle P. Congruent angles are angles that have the same measure. So, if angle G is congruent to angle P, it means that they have the same measure. For example, if angle G measures 30 degrees, then angle P would also measure 30 degrees.

Answer:

1. a. 25(1) + 10(2) + 3(3) + 10(4) = 94/140 = .671

2. b. Mark’s average < Jay’s average

Step-by-step explanation:

1. What is Mark’s slugging average?

The slugging average gives more weight to the multi-base hits (compared to single-base hits) in opposition with the batting average.

So, a single hit is worth 1 point, a double is worth 2 points, a triple 3 points and a homerun is worth 4 points.  It's a weighted average calculation.

Mark had 25 singles, 10 doubles, 3 triples and 10 homeruns during 140 presences 140 at bat.

25(1) + 10(2) + 3(3) + 10(4) = 94/140 = .671

2. How does Mark's average compares to Jay's?

Let's first calculate Jay's slugging average then we'll be able to decide.

Jay had 10 singles, 6 doubles, 5 triples and 14 homeruns in 90 presences.

10 (1) + 6 (2) + 5 (3) + 14 (4) = 10+12+15+56 = 93 / 90 = 1.033

We can definitely say that Mark's slugging average (0.671) calculated above is lower than Jay's average (1.033).  So,

b. Mark’s average < Jay’s average

In matrix form, we're looking for coefficients [tex]b_3,b_2,b_1,b_0[/tex] such that

[tex]\underbrace{\begin{bmatrix}3^3&3^2&3^1&3^0\\1^3&1^2&1^1&1^0\\(-1)^3&(-1)^2&(-1)^1&(-1)^0\\6^3&6^2&6^1&6^0\\7^3&7^2&7^1&7^0\end{bmatrix}}_{\mathbf A}\underbrace{\begin{bmatrix}b_3\\b_2\\b_1\\b_0\end{bmatrix}}_{\mathbf x}=\underbrace{\begin{bmatrix}4\\2\\1\\5\\9\end{bmatrix}}_{\mathbf b}[/tex]

The best-fit solution is given by [tex]\mathbf x=(\mathbf A^\top\mathbf A)^{-1}\mathbf A^\top\mathbf b[/tex]. You should end up with

[tex]\mathbf x=\begin{bmatrix}b_3\\b_2\\b_1\\b_0\end{bmatrix}\approx\begin{bmatrix}0.0495\\-0.3463\\0.9157\\2.1149\end{bmatrix}[/tex]

Attached is a plot of the given points and the best-fit solution.

To find the polynomial of degree three that best fits the given points, we can use the method of least squares.

To find the polynomial of degree three that best fits the given points, we can use the method of least squares. First, we can rewrite the equation in matrix form as:

[3 4 1] [b3]   [4]

[1 2 1] [b2] =  [2]

[-1 1 1] [b1]   [1]

[6 5 1] [b0]   [5]

[7 9 1]

Using matrix algebra, we can solve for the values of b3, b2, b1, and b0 that give us the best fit polynomial.

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Since the 5 is in the tens place, that means it represents 5 tens, or 50.

Answer:

the tens' place

Step-by-step explanation:

refer to the attachment

ANSWER

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EXPLANATION

We want to find the slope of a line that is perpendicular to the line y = 1

The line y=1 is parallel to the x-axis.

In other words, the line y=1 is a horizontal line.

The line perpendicular to y=1 is a vertical line.

The slope of a vertical line is undefined

Answer:

its an undefined slope

Step-by-step explanation:

the line y=−1 has slope 0 so any line perpendicular to it will have an undefined slope

Answer:

Number of adult tickets = 3 tickets

Number of children tickets = 5 tickets

Step-by-step explanation:

A- The system of equations:

Assume that the number of adult tickets is x and that the number of children tickets is y

We are given that:

i. The total number of people in the group is 8, which means that the total number of tickets bought is 8. This means that:

x + y = 8 ..................> equation I

ii. The price of an adult ticket is $12 and that of a child ticket is $10. We know that the group spent a total of $86. This means that:

12x + 10y = 86 ...............> equation II

From the above, the systems of equation is:

x + y = 8

12x + 10y = 86

B- Solving the system using elimination method:

Start by multiplying equation I by -10

This gives us:

-10x - 10y = -80 .................> equation III

Now, taking a look at equations II and III, we can note that coefficients of the y have equal values and different signs.

Therefore, we will add equations II and III to eliminate the y

    12x + 10y = 86

+( -10x - 10y = -80)

Adding the two equations, we get:

2x = 6

x = 3

Finally, substitute with x in equation I to get the value of y:

x + y = 8

3 + y = 8

y = 8 - 3 = 5

Based on the above:

Number of adult tickets = x = 3 tickets

Number of children tickets = y = 5 tickets

C- Explanation of the meaning of the solution:

The above solution means that for a group of 8 people to be able to spend $86 in a theater having the price of $12 for an adult ticket and $10 for a child' one, this group must be composed of 3 adults and 5 children

Hope this helps :)

Answer:

24cm and 225cm

Step-by-step explanation:

Answer:

1) [tex]24cm^2[/tex]

2) [tex]30cm[/tex]

Step-by-step explanation:

1) Remember that: The area of a quadrilateral circumscribed about a circle equals  semi-perimeter (half the product of the perimeter of the quadrilateral) and the radius of the circle.  

By the other hand in a quadrilateral circumscribed the sum of the measures of any pair of two opposite sides is equal to the sum of the measures of the other pair of the opposite sides.

Then you know that the sum of all sides should be 12cm+12cm=24cm

Area= Semi-perimeter*r

where

Semi-perimeter= perimeter/2= sum of all sides/2

r= radius of the circle

Then:

[tex]Area=(24cm/2)*(2cm)= 24cm^2[/tex]

2) Perimeter= sum of all sides

And as the sum of the measures of any pair of two opposite sides is equal to the sum of the measures of the other pair of the opposite sides.

Perimeter=15cm+15cm=30cm

Answer:

1/2

Step-by-step explanation:

sin(K)=5/10=1/2

The sine function in mathematics relates angles of a right triangle to the ratios of its side lengths. The value of sin(K) depends on the value of angle K, measured in radians.

In mathematics, the sine function is a trigonometric function that relates the angles of a right triangle to the ratios of the side lengths. The value of sin(K) depends on the value of angle K, which is measured in radians.

For example, if K = π/2, then sin(K) = 1. If K = 0, then sin(K) = 0.

To find the value of sin(K), you can use a calculator or reference table that provides the values of sine for different angles.

Answer: wheres the model?

Step-by-step explanation:

Answer:

[tex]P=25\%[/tex]

Step-by-step explanation:

We have a sample of 100 clients.

To find the probability that a randomly selected customer chooses a cold drink, we must first count how many people in the sample chose cold drinks.

The table shows that cold drinks were

8 small, 12 medium and 5 large.

Then the number of cold drinks was:

[tex]8 + 12 + 5 = 25[/tex]

Now the probability of someone selecting a cold drink is:

[tex]P = \frac{25}{100}[/tex]

[tex]P = 0.25 = 25\%[/tex]

Answer:p=25%

For the acellus people

Step-by-step explanation:

Answer:

2.74 seconds

Step-by-step explanation:

The apple will hit the ground when its height above the ground is zero:

0 = -16t^2 +120

t^2 = 120/16 . . . . . . divide by 16 and add t^2, then simplify and take the square root:

t = √7.5 ≈ 2.73861 ≈ 2.74 . . . . seconds

Answer:

x= 1.8

Step-by-step explanation:

We have been given the equation;

-(6)^(x-1)+5=(2/3)^(2-x)

We are required to determine the value of via graphing. To do this we can split up the right and the left hand sides of the equation to form the following two separate equations;

y = -(6)^(x-1)+5

y = (2/3)^(2-x)

We then proceed to graph the two equations on the same graph. The solution will be the point where the equations will intersect. Find the attachment below for the graph;

The value of x is 1.785. To the nearest tenth we have x = 1.8

Answer:

5

Step-by-step explanation:

We can use the distance formula with 3 different vertices to figure out the shortest of the three sides.

The distance formula is [tex]\sqrt{(y_2-y_1)^2+(x_2-x_1)^2}[/tex]

Where (x_1,y_1) is the first points and (x_2,y_2) is the second set of points, respectively.

Now let's figure out the length of 3 sides.

1. The length between (-6,-5) & (-5,6):

[tex]\sqrt{(y_2-y_1)^2+(x_2-x_1)^2} \\=\sqrt{(6--5)^2+(-5--6)^2}\\ =\sqrt{(6+5)^2+(-5+6)^2} \\=\sqrt{11^2+1^2} \\=\sqrt{122}[/tex]

2. The length between (-6,-5) & (-2,2):

[tex]\sqrt{(y_2-y_1)^2+(x_2-x_1)^2} \\=\sqrt{(2--5)^2+(-2--6)^2} \\=\sqrt{(2+5)^2+(-2+6)^2}\\ =\sqrt{7^2+4^2}\\ =\sqrt{65}[/tex]

3. The length between (-5,6) & (-2,2):

[tex]\sqrt{(y_2-y_1)^2+(x_2-x_1)^2} \\=\sqrt{(2-6)^2+(-2--5})^2}\\ =\sqrt{(-4)^2+(3)^2} \\=\sqrt{25} \\=5[/tex]

Thus, length of the shortest side is 5.

Answer:

5 units

Step-by-step explanation:

Draw the triangle with the vertices given. Identify the smallest side and draw in a right triangle using the smallest side of the original triangle as the hypotenuse of the right triangle, as shown on the grids below.

The horizontal side of the small right triangle measures 3 units and the vertical side of the small right triangle measures 4 units.

Use the Pythagorean theorem to find the length of the hypotenuse.

Therefore, the length of the shortest side of the triangle measures 5 units.

3^2 + 4^2  =  c^2

9 + 16  =  c^2

25  =  c^2

5  =  c

The answer is C. rectangular prism.

Hope this helps.

r3t40

iits probalty like based analyse

Answer:

A.Probability-based inference

Step-by-step explanation:

Inferential statistics uses a random sample of data from a population to draw inferences about the population. One can make generalizations about a population.

The answer as per scenario is A.Probability-based inference.

The manager made an inference based on probability.

Answer: 827 pounds.

Step-by-step explanation:

The formula for calculate density is:

[tex]d=\frac{m}{V}[/tex]

Where "m" is the mass and "V" is the volume.

The formula for calculate the volume of a cylinder is:

[tex]V=\pi r^2h[/tex]

Where "r" is the radius and "h" is the height.

Calculate the volume of the cylindrical maple tree stump, which is 3 feet high and has a radius of 1.5 feet:

[tex]V=(3.14)(1.5ft)^2(3ft)=21.195ft^3[/tex]

Substitute the volume into and the density 39 pounds per cubic foot, into the formula [tex]d=\frac{m}{V}[/tex] and solve for "m". Then, the weight of the stump to the nearest pound is:

[tex]39\frac{lb}{ft^3}=\frac{m}{21.195ft^3}\\\\m=(39\frac{lb}{ft^3})(21.195ft^3)\\\\m=827lb[/tex]

Answer:

14 years old

Explanation:

Use this system of equations if G=George and J=Jeannie:

G-12=J

50=(G+5)+(J+5)

So, change second equation becomes

40-G=J

Then use substitution:

G-12=40-G

2G=52

G=26

Then, to find Jeannie’s age, go back to one of the equations and substitute G.

26-12=J

So, J=14

Jeannie is currently 14 years old.

To solve any questions relating to ages, I find it easier to make a table, like in the picture below.

See picture for answer and clear diagram.

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

Notes/Explanations + answer in written form

Jennies age now = x

So Georges age now would be: x + 12   (since he is 12 years older)

Jennies age in 5 years would be: x + 5   (since it is 5 years later)

Georges age in 5 years would be: x + 12 + 5 = x + 17

We are told that their ages in 5 years would add up to 50. So we can write an equation like so:

x + 17   +   x + 5 = 50                 (All you have to do now is solve it)

2x + 22 = 50                             (combine like terms)

2x = 28                                      (subtract 22 from both sides)

x = 14                                         (divide both sides by two)

That means that Jennies age is 14 years

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

Note:

Please comment below if you have any questions - I would be more that happy to help / explain anything!

Answer:

g(x) = x^2 - 11x + 26

Step-by-step explanation:

In translation of functions, adding a constant to the domain values (x) of a function will move the graph to the left, while subtracting from the input of the function will move the graph to the right.

Given the function;

f(x) = x2 - 3x - 2

a shift 4 units to the right implies that we shall be subtracting the constant 4 from the x values of the function;

g(x) = f(x-4)

g(x) = (x - 4)^2 - 3(x - 4) -2

g(x) = x^2 - 8x + 16 - 3x + 12 - 2

g(x) = x^2 - 11x + 26

Answer:

g(x) = (x+4)^2 - 3x - 2

Step-by-step explanation:

A p e x

Question 1:

For this case we have that by definition, the GCF of two numbers is the largest number that is a factor of both numbers. For example, the number 10 is the biggest factor that 50 and 30 have in common.

We must find the GCF of the following expression:

[tex]30t ^ 2u + 12tu ^ 2 + 24tu[/tex]

We look for the prime factorization of each number:

[tex]30: 2 * 3 * 5\\12: 2 * 2 * 3\\24: 2 * 2 * 2 * 3[/tex]

The GCF of the three numbers is [tex]2 * 3 = 6[/tex]

So:

[tex]6tu (5t + 2u + 4)[/tex]

Answer:

Option C

Question 2:

For this case we have that by definition of power properties of the same base, that:

[tex]a ^ n * a ^ m = a ^ {n + m}[/tex]

That is, for multiplication of powers of the same base, the same base is placed and the exponents are added.

So:

[tex]b ^ n * b ^ m = b ^ {n + m}[/tex]

ANswer:

[tex]b ^ {n + m}[/tex]

Option A

The greatest common factor of the expression is 6tu, and the factored form is 6tu(5t + 2u + 4), . For the property of exponents, the rule is that when multiplying powers with the same base, you add the exponents, yielding an answer of  b^n+m.

15.The correct option is (C) and 16.The correct option is (A).

To solve the problem factor out the greatest common factor from the expression 30t2u + 12tu2 + 24tu, we have to look for the highest power of each variable and the largest number that can divide each term. The greatest common factor for this expression is 6tu.

Factoring out 6tu gives us:

The correct answer is C. 6tu(5t + 2u + 4).

For the property of exponents problem, when multiplying powers with the same base, you add the exponents. So, bn · bm = bn+m.

The correct answer is A. bn+m.

Answer:

g(x) = x^2 + 4x + 4

Step-by-step explanation:

In translation of functions, adding a constant to the domain values (x) of a function will move the graph to the left, while subtracting from the input of the function will move the graph to the right.

Given the function;

f(x) = x2 - 6x + 9

a shift 5 units to the left implies that we shall be adding the constant 5 to the x values of the function;

g(x) = f(x+5)

g(x) = (x+5)^2 - 6(x+5) + 9

g(x) = x^2 + 10x + 25 - 6x -30 + 9

g(x) = x^2 + 4x + 4

Step-by-step explanation:

First we need to find which function is the outside radius and which is the inside radius (by which I mean which is farther and which is closer to the axis of rotation).  We can do this by graphing, but we can also do this by evaluating each function at the end limits.

At x = 0:

y = 5 sin 0 = 0

y = 5 cos 0 = 5

At x = π/4:

y = 5 sin π/4 = 5√2 / 2

y = 5 cos π/4 = 5√2 / 2

So y = 5 cos x is the outside radius because it is farther away from y = -1, and y = 5 sin x is the inside radius because it is closer to y = -1.

Here's a graph:

desmos.com/calculator/5oaiobcpww

The volume of the rotation is:

V = π ∫ₐᵇ [(R−y)² − (r−y)²] dx

where a is the lower limit, b is the upper limit, R is the outside radius, r is the inside radius, and y is the axis of rotation.

Plugging in:

V = π ∫₀ᵖ [(5 cos x − -1)² − (5 sin x − -1)²] dx

V = π ∫₀ᵖ [(5 cos x + 1)² − (5 sin x + 1)²] dx

V = π ∫₀ᵖ [(25 cos² x + 10 cos x + 1) − (25 sin² x + 10 sin x + 1)] dx

V = π ∫₀ᵖ [25 cos² x + 10 cos x + 1 − 25 sin² x − 10 sin x − 1] dx

V = π ∫₀ᵖ [25 (cos² x − sin² x) + 10 cos x − 10 sin x] dx

V = π ∫₀ᵖ [25 cos (2x) + 10 cos x − 10 sin x] dx

V = π [25/2 sin (2x) + 10 sin x + 10 cos x] from 0 to π/4

V = π [25/2 sin (π/2) + 10 sin (π/4) + 10 cos (π/4)] − π [25/2 sin 0 + 10 sin 0 + 10 cos 0]

V = π [25/2 + 5√2 + 5√2] − 10π

V = π (5/2 + 10√2)

V ≈ 52.283

In this exercise it is necessary to calculate the volume of the rotating solid, in this way we have:

[tex]V= 52.3[/tex]

First we need to find which function is the outside radius and which is the inside radius (by which I mean which is farther and which is closer to the axis of rotation).  We can do this by graphing, but we can also do this by evaluating each function at the end limits.

[tex]x=0\\y=5sin(x)\\y=5sin(0)=0\\y=5cos(0)= 5\\\\x=\pi/4\\y=5sin(\pi/4)= 5(\sqrt{2/2})\\y=5cos( \pi/4)= 5/\sqrt{2}[/tex]

So [tex]y = 5 cos x[/tex] is the outside radius because it is farther away from y = -1, and [tex]y = 5 sin x[/tex] is the inside radius because it is closer to y = -1. Here's a graph (first image). The volume of the rotation is:

[tex]V=\pi\int\limits^a_b {[(R-y)^2-(r-y)^2]} \, dx[/tex]

Where a is the lower limit, b is the upper limit, R is the outside radius, r is the inside radius, and y is the axis of rotation. Plugging in:

[tex]V=\pi\int\limits^a_b {[(R-y)^2-(r-y)^2]} \, dx\\= \pi\int\limits^p_0 {[(5 cos(x) + 1)^2-(5 sin(x) + 1)^2]} \, dx\\=\pi\int\limits^p_0 {[(25 cos^2 (x) + 10 cos (x) + 1)-(25 sin^2 (x) + 10 sin (x) + 1)]} \, dx\\=\pi\int\limits^p_0 {[(5 cos^2 (x) + 10 cos (x) + 1 -25 sin^2 (x)- 10 sin (x)- 1]} \, dx\\=\pi\int\limits^p_0 {[(25 (cos^2( x)- sin^2( x)) + 10 cos( x)- 10 sin( x)]} \, dx\\=\pi\int\limits^p_0 {[(25 cos (2x) + 10 cos (x)- 10 sin (x)]} \, dx\\[/tex]

[tex]=\pi[(25/2 sin (2x) + 10 sin x + 10 cos (x)]\\\\=\pi[(25/2 sin (\pi/2) + 10 sin (\pi/4) + 10 cos (\pi/4)] - \pi [25/2 sin( 0) + 10 sin( 0) + 10 cos(0)]\\\\=\pi[25/2+5\sqrt{2}+5/\sqrt{2}]-10\pi\\\\\V= 52.3[/tex]

See more about volume of the solid at brainly.com/question/10956718

Answer:

240 feets

Step-by-step explanation:

In this question , apply Pythagorean relationship

Lets assume ;the climbing, sliding and walking back to the base forms a right-angle triangle

The tower height = the height of the triangle, h=60f

The sliding lane= the hypotenuse of the triangle=?

\

The distance covered to the base= base of the triangle=80f

Applying the relation ;

a²+b²=c²

80²+60²=c²

6400+3600=c²

10000=c²

√10000=c

100=c

Distance traveled in all=100+60+80=240 feet

Zachary traveled a total of 140 feet including climbing the tower and walking back from the splash pool to the tower.

The question is asking how far Zachary has traveled in total. He first climbs a 60-foot tower and then he lands in the splash pool. After that, he walks 80 feet back to the base of the tower.

To find the total distance he traveled, we need to add the distance he climbed up the tower to the distance he walked back to the tower. This can be done using simple addition.

Therefore, the total distance= distance climbed + distance walked = 60 feet + 80 feet = 140 feet.

So, Zachary traveled a total of 140 feet in all.

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B can be right but if he dont do the exercises it is wrong

so it is C because d=day so it is 15 day and the results of the function is calories he used to do the exercises so the answer is C

Answer:

  C is the correct answer

Step-by-step explanation:

There is no "work." The question is a reading comprehension question, so the work goes on in your brain, where the words are interpreted. The problem statement tells you ...

  "... the amount of calories he has used, f(d), is a function of the number of days, d, he has exercised with the new routine."

You are comparing this wording to the answer choice wording. The best match is to C (as you know), which says ...

  "f(15) = 5250 means after 15 days of exercising with his new routine, Vincent has used 5250 calories."

Answer:

Step-by-step explanation:

The ratio of interest rates is 8% : 4% = 2 : 1. Since the amount of interest earned in each account is the same, the ratio of amounts invested will be the inverse of that, 1/2 : 1/1 = 1 : 2.

Then 1/(1+2) = 1/3 of the money is invested in the 8% account. That amount is ...

  $9102/3 = $3034 . . . . invested at 8%

The remaining amount is invested at 4%:

  $9102 - 3034 = $6068 . . . . invested at 4%

_____

The interest earned in 1 year in each account is

  0.08·$3034 = 0.04·$6068 = $242.72

_____

If you really need an equation, you can let x represent the amount invested at the higher rate. (Using this variable assignment avoids negative numbers later.) Then 9102-x is the amount invested at the lower rate.

  0.08x = 0.04(9102-x)

  0.12x = 0.04·9102 . . . . . eliminate parentheses, add 0.04x

  x = (0.04/0.12)·9102 = 9102/3 . . . . . divide by the coefficient of x. Same answer as above.

ANSWER

[tex]{x}^{3} - {y}^{3} = (x - y)[ {x}^{2} + xy + {y}^{2}][/tex]

EXPLANATION

We want to factor:

[tex] {x}^{3} - {y}^{3} [/tex]

completely.

Recall from binomial theorem that:

[tex]( {x - y)}^{3} = {x}^{3} - 3 {x}^{2} y + 3x {y}^{2} - {y}^{3} [/tex]

We make x³-y³ the subject to get:

[tex] {x}^{3} - {y}^{3} = ( {x - y)}^{3} + 3 {x}^{2} y - \:3x {y}^{2}[/tex]

We now factor the right hand side to get;

[tex]{x}^{3} - {y}^{3} = ( {x - y)}^{3} + 3 {x} y(x - y)[/tex]

We factor further to get,

[tex]{x}^{3} - {y}^{3} = (x - y)[( {x - y)}^{2} + 3 {x} y][/tex]

[tex]{x}^{3} - {y}^{3} = (x - y)[ {x}^{2} - 2xy + {y}^{2} + 3 {x} y][/tex]

This finally simplifies to:

[tex]{x}^{3} - {y}^{3} = (x - y)[ {x}^{2} + xy + {y}^{2}][/tex]

To factor the expression x³ - y³ completely, use the formula for factoring a difference of cubes: a³ - b³ = (a - b)(a² + ab + b²). Plug in x for a and y for b to get (x - y)(x² + xy + y²).

To factor the expression x³ - y³ completely, we can use the formula for factoring a difference of cubes. The formula is:

a³ - b³ = (a - b)(a² + ab + b²)

Using this formula, we can plug in x for a and y for b.

x³ - y³ = (x - y)(x² + xy + y²)

Therefore, the expression x³ - y³ can be factored completely as (x - y)(x² + xy + y²).

Answer:

π/49·(44 -π)

Step-by-step explanation:

Using the "shell" method, a differential of volume is the product of the area of a cylindrical shell and its circumference around the axis of rotation. Here, that area is ...

dA = y·dx = cos(7x)dx

The radius will be the difference between x and the axis of rotation, x=3, so is ...

r = 3 -x

Then the differential of volume is ...

dV = 2πr·dA = 2π(3-x)cos(7x)dx

The volume will be the integral of this over the limits x ∈ [0, π/14].

∫dV = 6π·∫cos(7x)dx -2π·∫x·cos(7x)dx . . . from 0 to π/14

= (6/7)π·sin(7x) -(2/49)π·(cos(7x) +7x·sin(7x)) . . . from 0 to π/14

= (6/7)π(1 -0) -(2/49)π((0 -1) +(π/2-0))

= π(42/49 +2/49 -π/49)

= (π/49)(44 -π) . . . . cubic units . . . . . approx 2.61960 cubic units

Final answer:

The volume of the solid created by rotating the bounded region about the axis x=3 is found using cylindrical shells and evaluating the integral from x=0 to x=π/14 of the expression 2π(3-x)cos(7x)dx.

Explanation:

To find the volume of the solid obtained by rotating the region bounded by the curves y=0, y=cos(7x), x=π/14, and x=0 about the axis x=3, we need to use the method of cylindrical shells.

For a small strip at position x with height y and thickness dx, rotating around x=3 creates a cylindrical shell with circumference 2π(3-x), height cos(7x), and thickness dx.

The volume of each shell is V = 2π(3-x)cos(7x)dx, and summing these from x=0 to x=π/14 gives us the total volume.

So, the volume V is the integral:

∫0π/14 2π(3-x)cos(7x)dx.

This volume is found by evaluating the definite integral.

Answer:

x = 3

Step-by-step explanation:

3/2.25 = 4/x                 Cross multiply

3x = 2.25 * 4                Combine the right

3x = 9                           Divide by 3

3x/3 = 9/3

x = 3

Answer:

D

Step-by-step explanation:

[tex]\left[\begin{array}{cc}Flow Rate&Eroded Soil\\0.31&0.82\\0.85&1.95\\1.26&2.18\\2.47&3.01\\3.75&6.07\end{array}\right][/tex]

As we can see, as flow rate increases, eroded soil also increases.  So the association is positive.

The association between flow rate and amount of eroded soil is positive.

The association between flow rate and amount of eroded soil is:

D. positive

The data shows that as the water flow rate increases, the amount of eroded soil also increases. This positive relationship indicates that higher water flow leads to more soil erosion.

Answer:

[tex]127,000\ members[/tex]

Step-by-step explanation:

In this problem we have an exponential function of the form

[tex]f(x)=a(b)^{x}[/tex]

where

a is the initial value

b is the base

The base is equal to

b=1+r

r is the average rate

In this problem we have

a=23,000 members

r=5%=5/100=0.05

b=1+0.05=1.05

substitute

[tex]f(x)=23,000(1.05)^{x}[/tex]

x ----> is the number of years since 1985

How many members will there be in 2020?

x=2020-1985=35 years

substitute in the function

[tex]f(x)=23,000(1.05)^{35}=126,868\ members[/tex]

Round to the nearest thousand

[tex]126,868=127,000\ members[/tex]

The club is expected to have about 126868 members in 2020.

[tex]\[N(t) = N_0 \times (1 + r)^t\][/tex]

In this problem:

[tex]- \( N_0 = 23,000 \)- \( r = 0.05 \) (5% growth rate)- \( t = 2020 - 1985 = 35 \)[/tex]

Substituting these values into the formula, we get:

[tex]\[N(35) = 23,000 \times (1 + 0.05)^{35}\][/tex]

[tex]\[N(35) = 23,000 \times (1.05)^{35}\][/tex]

[tex]\[(1.05)^{35} \approx 5.516\][/tex]

Now, multiply by the initial number of members:

[tex]\[N(35) = 23,000 \times 5.516 \approx 126,868\][/tex]