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(a) The volume of the solid is 96 cubic units.

(b) The volume of the solid is (160/5) cubic units.

a) Since the cross-sections are rectangles of height 8 and the base lies in the xy-plane, the area of each cross-section is the product of the height and the length along the x-axis, which is given by the difference between the y-coordinates of the two lines y=6x and y=12. So the area of each cross-section is,

A(x) = 8(12-6x) = 96 - 48x

To find the volume of the solid, we need to integrate the area function A(x) from x=0 to x=2 (the x-coordinate of the intersection between y=6x and y=12):

V = ∫₀² A(x) dx

= ∫₀² (96 - 48x) dx

= [96x - 24x²] from x=0 to x=2

= (192 - 96) = 96

b) Since the cross-sections are rectangles of perimeter 60, we know that the width and height of each rectangle are related by the equation 2w + 2h = 60, or equivalently, w + h = 30. Also, the width of each rectangle is the difference between the y-coordinates of the two lines y=6x and y=12, so we have w = 12 - 6x.

Substituting for w in terms of x in the equation w + h = 30, we get:

(12 - 6x) + h = 30

h = 18 - 12x

Therefore, the area of each cross-section is,

A(x) = w*h = (12-6x)(18-12x) = 216 - 216x + 72x²

To find the volume of the solid, we need to integrate the area function A(x) from x=0 to x=2:

V = ∫₀² A(x) dx

= ∫₀² (216 - 216x + 72x²) dx

= [216x - 108x² + 24x³] from x=0 to x=2

= (432/3) - (864/4) + (192/5)

= 160/5

which can be simplified as 32√5/5 cubic units.

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Answer:

A

Step-by-step explanation:

The maximum and minimum values of the function is solved

Given data ,

We can find the maximum and minimum values of the function by taking the derivative of y with respect to x and setting it equal to zero.

y = (sin x)³ - (sin x)⁶

y' = 3(sin x)² cos x - 6(sin x)⁵ cos x

Setting y' equal to zero:

0 = 3(sin x)² cos x - 6(sin x)⁵ cos x

0 = 3(sin x)² cos x (1 - 2(sin x)³)

sin x = 0 or (sin x)³ = 1/2

If sin x = 0, then x = kπ for any integer k.

If (sin x)³ = 1/2, then sin x = (1/2)^(1/3) ≈ 0.866. This occurs when x = π/3 + 2kπ/3 or x = 5π/3 + 2kπ/3 for any integer k.

To determine whether these values correspond to a maximum or minimum, we can use the second derivative test.

y'' = 6(sin x)³ cos² x - 15(sin x)⁴ cos² x - 9(sin x)⁴ cos x + 6(sin x)⁵ cos x

y'' = 3(sin x)³ cos x (4(sin x)² - 5(sin x)² - 3cos x + 2)

For x = kπ, y'' = 3(0)(-3cos(kπ) + 2) = 6 or -6, depending on the parity of k. This means that these points correspond to a maximum or minimum, respectively.

For x = π/3 + 2kπ/3 or x = 5π/3 + 2kπ/3, y'' = 3(1/2)^(5/3) cos x (4(1/2)^(2/3) - 5(1/2)^(1/3) - 3cos x + 2). This expression is positive for x = π/3 + 2kπ/3 and negative for x = 5π/3 + 2kπ/3, which means that the former correspond to a minimum and the latter to a maximum.

Hence , the maximum value of the function is y = 27/64, which occurs at x = 5π/3 + 2kπ/3, and the minimum value is y = -1/64, which occurs at x = π/3 + 2kπ/3

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Answer:

Step-by-step explanation:

You want the maximum and minimum values of the function ...

  y = sin³(x) -sin⁶(x)

When we substitute sin³(x) = z, we have the quadratic expression ...

  y = z -z² . . . . . a quadratic function

Adding and subtracting 1/4, we can put this in vertex form:

  y = -(z -1/2)² +1/4

This version of the function describes a parabola that opens downward and has a vertex at (z, y) = (1/2, 1/4). The y-value of the vertex represents the maximum value of the function.

The maximum value of y is 1/4.

The sine function is a continuous function with a range of [-1, 1]. Then z will be a continuous function of x, with a similar range. We already know that y describes a function of z that is a parabola opening downward with a line of symmetry at z = 1/2. This means the most negative value of y will be found at z = -1 (the value of z farthest from the line of symmetry). That value of y is ...

  y = (-1) -(-1)² = -1 -1 = -2

The minimum value of y is -2.

__

Additional comment

The range of y is confirmed by a graphing calculator.

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Answer:

pi

Step-by-step explanation:

Step-by-step explanation:

Using Quadratic Formula   with   a = 1     b= 4       c = 8

roots are  x = -2 - 2i      and   x = -2 + 2i

We can integrate this S = 2π ∫(1.1 to 4.4) (5√(4x + 25))/(2√x) dx over the given interval (1.1 to 4.4) to find the surface area.

We can evaluate the integral using numerical methods or a calculator to find the final answer.

We have,

To find the surface area of the revolution about the x-axis of the function f(x) = 5√x over the interval (1.1 to 4.4), we can use the formula for the surface area of revolution:

S = ∫(a to b) 2πy√(1 + (f'(x))²) dx

In this case,

f(x) = 5√x, so f'(x) = (d/dx)(5√x) = 5/(2√x).

Let's calculate the surface area:

S = ∫(1.1 to 4.4) 2π(5√x)√(1 + (5/(2√x)²) dx

Simplifying the expression inside the integral:

S = ∫(1.1 to 4.4) x 2π(5√x)√(1 + 25/(4x)) dx

Next, we can integrate this expression over the given interval (1.1 to 4.4) to find the surface area.

To find the surface area of revolution about the x-axis of the function

f(x) = 5√x over the interval (1.1 to 4.4), we need to evaluate the integral:

S = ∫(1.1 to 4.4) 2π(5√x)√(1 + 25/(4x)) dx

Let's calculate the integral:

S = 2π ∫(1.1 to 4.4) (5√x)√(1 + 25/(4x)) dx

To simplify the calculation, let's simplify the expression inside the integral first:

S = 2π ∫(1.1 to 4.4) (5√x)√((4x + 25)/(4x)) dx

Next, we can distribute the square root and simplify further:

S = 2π ∫(1.1 to 4.4) (5√(4x + 25))/(2√x) dx

Thus,

We can integrate this expression over the given interval (1.1 to 4.4) to find the surface area.

We can evaluate the integral using numerical methods or a calculator to find the final answer.

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The probability of more than 3 successes in 6 trials with a probability of success of 0.6 is 0.4558.

To calculate the probability of more than 3 successes in 6 independent trials with a binomial distribution where the probability of success is 0.6, we can use the cumulative distribution function (CDF) of the binomial distribution:

Pr(x > 3) = 1 - Pr(x ≤ 3)

To find Pr(x ≤ 3), we can use the binomial probability mass function (PMF) for each possible value of x and add them up:

Pr(x = 0) = (6 choose 0) × 0.6^0 × 0.4^6 = 0.0041

Pr(x = 1) = (6 choose 1) × 0.6^1 × 0.4^5 = 0.0409

Pr(x = 2) = (6 choose 2) × 0.6^2 × 0.4^4 = 0.1536

Pr(x = 3) = (6 choose 3) × 0.6^3 × 0.4^3 = 0.3456

Therefore, Pr(x ≤ 3) = 0.0041 + 0.0409 + 0.1536 + 0.3456 = 0.5442

Finally, we can calculate Pr(x > 3) as follows:

Pr(x > 3) = 1 - Pr(x ≤ 3) = 1 - 0.5442 = 0.4558

Therefore, the probability is 0.4558

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To find the height of the cylinder when the volume is given as 1500 in³ and the radius is 7 inches, we can use the formula for the volume of a cylinder:

Volume = π * r² * h

Substituting the given values, we have:

\(1500 = 3.14 * 7^2 * h1500 = 3.14 * 49 * h1500 = 153.86 * h\)

To solve for h, we divide both sides of the equation by 153.86:

h = 1500 / 153.86

h ≈ 9.75

Rounding the answer to the nearest hundredth, the height of the cylinder is approximately 9.75 inches.

Therefore, the height of the cylinder is 9.75 inches.

Note: It is important to use the accurate value of π, which is approximately 3.14159, for precise calculations. However, in this case, since you specified to use 3.14 for π, I have used that approximation to calculate the height.

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Answer:

-2x

Step-by-step explanation:

As you can see the line is perpendicular the the given equation and passes through the point (-1,2).

Answer:

y=3x+5

Step-by-step explanation:

the equation x+3y=3 is in standard form (ax+by=c)

we can put it into slope intercept form (y=mx+b) to make it easier.

subtract x from both sides

3y=-x+3

divide by 3

y=-1/3x+1

Perpendicular lines have slopes that are opposite (one is negative and one is positive) and reciprocal, so if you mutiply them together, it will equal -1

use this equation:

-1/3m=-1

multiply by -3 on both sides

m=3

equation so far: y=3x+b (b is a placeholder)

since the equation passes through the point (-1,2), we can substitute it into the equation

2=3(-1)+b

2=-3+b

5=b

the equation is y=3x+5

Hope this helps!

Answer:

30

Step-by-step explanation:

(n+8)=2(7+8)=2(15)=30

Answer:

30

Step-by-step explanation:

The Subtraction property of equality justifies the conclusion based on the given information.

The subtraction property of equality states that when the same number is subtracted from both sides of an equality, then the two sides of the equation still remain equal. In simple words, we can say that when a number is subtracted from one side of an equation, the same number must be subtracted from the other side of the equality.

if a = b

then;

a-c = b-c

We have:

If x + 5 = 15

By subtraction property of equality:

Subtract 5 from both sides we have;

x = 10

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Answer:

1,133

Step-by-step explanation:

2%=0.02

$22.66÷0.02=$1,133

Answer:

ATLA, or pokemon.

Step-by-step explanation:

You never said Japanese anime so I hope you enjoy American style anime. (Or pokemon)

Answer:

demon slayer

death note

hunter x hunter

bleach

my hero academia

dxd ( if your into s3xual stuff)

naruto

Step-by-step explanation:

ive watched all of those and they are pretty good:)

Brainliest please<33

Answer:

4.27 x \(10^{5}\)

Step-by-step explanation:

The whole digit cannot go past 10 or be less than 1:

1 - 10 is the numbers of its max.

427,000

We would move the decimal point between 4 and 2 making:

4.2700

(Originally the decimal point moved 5 places because 427,000 is a whole number, which means the decimal point would be considered at the end, as shown below)

427,000.

By moving 5 places to the left this will be part of your: scientific notation as

___ x \(10^{5}\)

(Then plug in 4.27 in front of your notation)

4.27 x \(10^{5}\)

That is your answer!

Answer:5SQRT61/

61

Step-by-step explanation:

(a) When measuring angle 9 and the lengths of AB and BC, one observation is required.

(b) The line drawn from the center of a circle to the midpoint of a chord is perpendicular to the chord, while the line drawn from the center of a circle to an endpoint of a chord bisects the chord.

a) In the given diagram, we have angle AOB (angle 9) and line segments AB and BC.

To measure angle 9, we use a protractor. We align the baseline of the protractor with the line segment OA, ensuring that the center point of the protractor coincides with the vertex O. Then, we read the degree measurement where the line segment OB intersects the protractor. Let's assume the measurement of angle 9 is x degrees.

To measure the length of line segments AB and BC, we use a ruler. We align the ruler with the line segments and read the length in units (e.g., centimeters or inches).

b) The statements to complete are:

(i) The line drawn from the center of a circle to the midpoint of a chord will be perpendicular to the chord.

(ii) The line drawn from the center of a circle to the endpoint of a chord will bisect the chord.

In a circle, the center of the circle lies on the perpendicular bisector of any chord. This means that if we draw a line from the center of the circle to the midpoint of a chord, it will be perpendicular to the chord (statement i). Similarly, if we draw a line from the center of the circle to one of the endpoints of the chord, it will bisect the chord into two equal parts (statement ii).

These properties hold true for any chord in a circle, and they are based on the geometric properties of circles and perpendicular bisectors.

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Here is the final matching of multiplication expressions with their estimated products:

(80)(30) = 2,400

(30)(6) = 180

(30)(5) = 150

(60)(4) = 240

(50)(30) = 1,500

29.3 x 5.9 ≈ 173

81.4 x 32.1 ≈ 2,618

32.9 x 4.81 ≈ 158

46.7 x 31.7 ≈ 1,479

59.3 x 3.57 ≈ 212

Match each multiplication expression on the left with the best estimate of its product on the right:

(80)(30) = 2,400

(30)(6) = 180

(30)(5) = 150

(60)(4) = 240

(50)(30) = 1,500

29.3 x 5.9

81.4 x 32.1

32.9 x 4.81

46.7 x 31.7

59.3 x 3.57

Matching the expressions with their estimated products:

(80)(30) = 2,400

(30)(6) = 180

(30)(5) = 150

(60)(4) = 240

(50)(30) = 1,500

Estimates for the remaining expressions:

29.3 x 5.9 ≈ 173.27

81.4 x 32.1 ≈ 2,612.94

32.9 x 4.81 ≈ 158.05

46.7 x 31.7 ≈ 1,480.39

59.3 x 3.57 ≈ 211.46

Matching the expressions with their estimated products:

(80)(30) = 2,400

(30)(6) = 180

(30)(5) = 150

(60)(4) = 240

(50)(30) = 1,500

29.3 x 5.9 ≈ 173.27

81.4 x 32.1 ≈ 2,612.94

32.9 x 4.81 ≈ 158.05

46.7 x 31.7 ≈ 1,480.39

59.3 x 3.57 ≈ 211.46

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A common aspect ratio for computer screens is 16:9, which means that for every 16 units of width, there are 9 units of height. This aspect ratio is also used on many televisions and mobile devices.

If a screen has an aspect ratio of 16:9, the width and height will be similar on all screens with this aspect ratio. However, the diagonal measurement will vary depending on the size of the screen.

Congruent screens are two or more screens that have the same dimensions. Two screens are congruent if and only if corresponding sides are equal and corresponding angles are equal. This can be proven using the postulates of congruence, such as the SSS (Side-Side-Side) postulate or the SAS (Side-Angle-Side) postulate.

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Answer:

What are the coordinates of point P after a counterclockwise rotation of 90°?

Step-by-step explanation:

First find the area of the triangle using the formula 1/2 x base x height:

1/2 x 22 x 13 = 143 square cm

Now for volume multiply the area by the depth of the figure:

143 x 30 = 4,290 cubic cm

Answer:

780

Step-by-step explanation:

First use pythagorean theorem to find the triangle, then multiply by 30

Answer: h=150

Step-by-step explanation:

2h/15 = 20

     2h= 20*15 [multiply 15 in both side]

     2h= 300

       h= 300/2

       h=  150

Desired expression for the polar arc length is L = ∫\(\sqrt{(r^2 + r'\:^2)}\)dθ

To make the substitution x=rcos(θ) and y=rsin(θ) for a polar function r=f(θ), we need to express dx/dθ and dy/dθ in terms of r and dr/dθ.

We have:

dx/dθ = -rsin(θ) + r'cos(θ)

dy/dθ = rcos(θ) + r'sin(θ)

where r' denotes the derivative of r with respect to θ, i.e., dr/dθ.

We can use these expressions to simplify the integrand in the expression for arc length:

\(\sqrt{(dx/d\theta)^2}\) + \((dy/d\theta)^2\)

= \(\sqrt{(-rsin(\theta) + r'cos(\theta)^2} + (rcos(\theta) + r'sin(\theta))^2)\)

= \(\sqrt{(r^2sin^2(\theta) - 2rr'sin(\theta)cos(\theta) + r'^2cos^2(\theta) + r^2cos^2(\theta) + 2rr'sin(\theta)cos(\theta) + r'^2sin^2(\theta)}\)

= \(\sqrt{(r^2 + r'\:^2)}\)

where we have used the identity \(sin^2\theta + cos^2 \theta\) = 1 to simplify the terms involving sin(θ)cos(θ).

Substituting this expression into the integral for arc length, we have:

L = ∫\(\sqrt{(r^2 + r'\:^2)}\)dθ

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Explanation:

Malise has 1 something (may be a pound, any unit) and she uses 1/8 to make pancakes and 1/3 to make a pie. To see what fraction of blueberries are left we have to substract 1/8 and 1/3 from 1:

We have to find the lower common denominator. Since 8has factors 2, 2, 2 and 3 has itself as a factor (and of course 1 for both) the LCD is the product of these two: 8 x 3 = 24.

Now we have to write it and in the numerator, we have to write each numerator multiplied by the quotient between the LCD and each denominator:

Answer:

The fraction of blueberries left is 13/24

Answer: 1 /8 - 1/3

Step-by-step explanation:

8 x 3 = 24

Answer:

x=10

Step-by-step explanation:

divide by 3 on both sides then add 6 on both sides

Answer:

the answer is x=10, x=2

Step-by-step explanation:

hope that helps

Answer:

the answer is to this question is -25

Answer:

1) -4

2) -18

3) -25

Step-by-step explanation:

1)

-6 + -8 = -14

-14 + 10 = -4

2)

Any number multiplied by a negative number will result in a negative product.

So what I do is I multiply as if there was no negative sign: 3 × 6 = 18

Then I add the negative sign to the product: -18

3)

One way you could do this is by switching it around: 18 - -7

Which would make it: 18 + 7 = 25

Then add the negative sign: -25

First you have to calculate the missing angle from the triangle AHI.

To do so you have to remember that the sum of all internal angles of a triengle equals 180º

m∠HAI+m∠AHI+m∠AIH=180º

59º+61º+m∠AIH=180º

m∠AIH=180º-59º-61º

m∠AIH=60º

Next, angles m∠AIH and m∠AKJ are corresponding angles, this means that both angles are congruent (equal), so

m∠AIH=m∠AKJ → replace

60º=xº