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Tom [10]
2 years ago
15

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves

about the x-axis. (tried this question a lot and I am just missing something)(plz help)
x = −3y2 + 9y − 6, x = 0
Mathematics
1 answer:
Vadim26 [7]2 years ago
6 0

The expression on the left side describes a parabola. Factorize it to determine where it crosses the y-axis (i.e. the line x = 0) :

-3y² + 9y - 6 = -3 (y² - 3y + 2)

… = -3 (y - 1) (y - 2) = 0

⇒   y = 1   or   y = 2

Also, complete the square to determine the vertex of the parabola:

-3y² + 9y - 6 = -3 (y² - 3y) - 6

… = -3 (y² - 3y + 9/4 - 9/4) - 6

… = -3 (y² - 2•3/2 y + (3/2)²) + 27/4 - 6

… = -3 (y - 3/2)² + 3/4

⇒   vertex at (x, y) = (3/4, 3/2)

I've attached a sketch of the curve along with one of the shells that make up the solid. For some value of x in the interval 0 ≤ x ≤ 3/4, each cylindrical shell has

radius = x

height = y⁺ - y⁻

where y⁺ refers to the half of the parabola above the line y = 3/2, and y⁻ is the lower half. These halves are functions of x that we obtain from its equation by solving for y :

x = -3y² + 9y - 6

x = -3 (y - 3/2)² + 3/4

x - 3/4 = -3 (y - 3/2)²

-x/3 + 1/4 = (y -  3/2)²

± √(1/4 - x/3) = y - 3/2

y = 3/2 ± √(1/4 - x/3)

y⁺ and y⁻ are the solutions with the positive and negative square roots, respectively, so each shell has height

(3/2 + √(1/4 - x/3)) - (3/2 - √(1/4 - x/3)) = 2 √(1/4 - x/3)

Now set up the integral and compute the volume.

\displaystyle 2\pi \int_{x=0}^{x=3/4} 2x \sqrt{\frac14 - \frac x3} \, dx

Substitute u = 1/4 - x/3, so x = 3/4 - 3u and dx = -3 du.

\displaystyle 2\pi \int_{u=1/4-0/3}^{u=1/4-(3/4)/3} 2\left(\frac34 - 3u\right) \sqrt{u} \left(-3 \, du\right)

\displaystyle -12\pi \int_{u=1/4}^{u=0} \left(\frac34 - 3u\right) \sqrt{u} \, du

\displaystyle 12\pi \int_{u=0}^{u=1/4} \left(\frac34 u^{1/2} - 3u^{3/2}\right)  \, du

\displaystyle 12\pi \left(\frac34\cdot\frac23 u^{3/2} - 3\cdot\frac25u^{5/2}\right)  \bigg|_{u=0}^{u=1/4}

\displaystyle 12\pi \left(\frac12 u^{3/2} - \frac65u^{5/2}\right)  \bigg|_{u=0}^{u=1/4}

\displaystyle 12\pi \left(\frac12 \left(\frac14\right)^{3/2} - \frac65\left(\frac14\right)^{5/2}\right) - 12\pi (0 - 0)

\displaystyle 12\pi \left(\frac1{16} - \frac3{80}\right) = \frac{12\pi}{40} = \boxed{\frac{3\pi}{10}}

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Hitman42 [59]

Answer:

Log [x^2 / yz^2]

Step-by-step explanation:

2log(x) =log(x^2)

-log(y) = The minus sign means that you can divide it by first log(x^2)

-2log(z) =log(z^2), and the minus sign again means you can divide it by the first log(x^2)

So, the whole thing can be written like this:

Log(x^2) / [log(y)*log(z^2)], OR:

Log [x^2 / yz^2]

4 0
3 years ago
If you are offered one slice from a round pizza (in other words, a sector of a circle) and the slice must have a perimeter of 24
VARVARA [1.3K]

Answer:

A 12 in diameter will reward you with the largest slice of pizza.

Step-by-step explanation:

Let r be the radius and \theta be the angle of a circle.

According with the graph, the area of the sector is given by

A=\frac{1}{2}r^2\theta

The arc lenght of a circle with radius r and angle \theta is r \theta

The perimeter of the pizza slice is composed of two straight pieces, each of length r inches, and an arc of the circle which you know has length s = rθ inches. Thus the perimeter has length

2r+r\theta=24 \:in

We need to express the area as a function of one variable, to do this we use the above equation and we solve for \theta

2r+r\theta=24\\r\theta=24-2r\\\theta=\frac{24-2r}{r}

Next, we substitute this equation into the area equation

A=\frac{1}{2}r^2(\frac{24-2r}{r})\\A=\frac{1}{2}r(24-2r)\\A=12r-r^2

The domain of the area is

0

To find the diameter of pizza that will reward you with the largest slice you need to find the derivative of the area and set it equal to zero to find the critical points.

\frac{d}{dr} A=\frac{d}{dr}(12r-r^2)\\A'(r)=\frac{d}{dr}(12r)-\frac{d}{dr}(r^2)\\A'(r)=12-2r

12-2r=0\\-2r=-12\\\frac{-2r}{-2}=\frac{-12}{-2}\\r=6

To check if r=6 is a maximum we use the Second Derivative test

if f'(c)=0 and f''(c), then f(x) has a local maximum at x = c.

The second derivative is

\frac{d}{dr} A'(r)=\frac{d}{dr} (12-2r)\\A''(r)=-2

Because A''(r)=-2 the largest slice is when r = 6 in.

The diameter of the pizza is given by

D=2r=2\cdot 6=12 \:in

A 12 in diameter will reward you with the largest slice of pizza.

3 0
3 years ago
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Brut [27]
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4 0
2 years ago
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andrey2020 [161]
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(x^4+xy^3)~~+~~(x^3+y^3)\implies x(x^3+y^3)~~+~~(x^3+y^3)
\\\\\\
\stackrel{\stackrel{common}{factor}}{(x^3+y^3)}(x+1)\\\\
-------------------------------\\\\
\textit{now recall that }\qquad \qquad \textit{difference of cubes}
\\ \quad \\
a^3+b^3 = (a+b)(a^2-ab+b^2)\qquad
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5 0
3 years ago
16 square meters is equivalent o how many square yards
Marat540 [252]

Answer: 19.13 yd²

Step-by-step explanation:

1. By definition,  you have that 1 square meter is equal to 1.19599 square yards. You can express it as following:

1 m²= 1.19599 yd²

2. Then, keeping the information above on mind, you can make the conversion from 16 m² to yd² as it is shown below:

(16m^2)(\frac{1.19599yd^2}{1m^2})=19.13yd^2

Therefore, you have that 16 m² is equivalent to 19.13 yd².

6 0
3 years ago
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