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Fittoniya [83]
2 years ago
14

Find the volume of the solid obtained by rotating the region bounded by y=4x and y=2sqrt(x) about the line x=6.

Mathematics
1 answer:
Pie2 years ago
7 0

Check the picture below.

so by graphing those two, we get that little section in gray as you see there, now, x = 6  is a vertical line, so we'll have to put the equations in y-terms and this is a washer, so we'll use the washer method.

y=4x\implies \cfrac{y}{4}=x\qquad \qquad y=2\sqrt{x}\implies \cfrac{y^2}{4}=x~\hfill \begin{cases} \cfrac{y}{4}=x\\\\ \cfrac{y^2}{4}=x \end{cases}

the way I get the radii is by using the "area under the curve" way, namely, I use it to get R² once and again to get r² and using each time the axis of rotation as one of my functions, in this case the axis of rotation will be f(x), and to get R² will use the "farthest from the axis of rotation" radius, and for r² the "closest to the axis of rotation".

\stackrel{R}{\stackrel{f(x)}{6}-\stackrel{g(x)}{\cfrac{y^2}{4}}}\qquad \qquad \stackrel{r}{\stackrel{f(x)}{6}-\stackrel{g(x)}{\cfrac{y}{4}}}~\hfill \stackrel{R^2}{\left( 6-\cfrac{y^2}{4} \right)^2}-\stackrel{r^2}{\left( 6-\cfrac{y}{4} \right)^2} \\\\\\ \stackrel{\textit{doing a binomial expansion and simplification}}{3y-3y^2-\cfrac{y^2}{16}+\cfrac{y^4}{16}}

now, both lines if do an equation on where they meet or where one equals the other, we'd get the values for y = 0 and y = 1, not surprisingly in the picture.

\displaystyle\pi \int_0^1\left( 3y-3y^2-\cfrac{y^2}{16}+\cfrac{y^4}{16} \right)dy\implies \pi \left( \left. \cfrac{3y^2}{2} \right]_0^1-\left. y^3\cfrac{}{} \right]_0^1-\left. \cfrac{y^3}{48}\right]_0^1+\left. \cfrac{y^5}{80} \right]_0^1 \right) \\\\[-0.35em] ~\dotfill\\\\ ~\hfill \cfrac{59\pi }{120}~\hfill

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6 0
3 years ago
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g100num [7]
I think the pattern is that you multiply the first number by the second , then add the first number. SO:
e.g. for the first one 1 x 4 = 4 , 4 + 1 = 5
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7 0
3 years ago
Find the standard deviation for the set of data. (17,26,28,9,30, 29,6,21,23}​
gogolik [260]

Using it's concept, the standard deviation for the given data-set is of 8.22.

<h3>What are the mean and the standard deviation of a data-set?</h3>

  • The mean of a data-set is given by the <u>sum of all values in the data-set, divided by the number of values</u>.
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The mean for this problem is:

M = (17 + 26 + 28 + 9 + 30 + 29 + 6 + 21 + 23)/9 = 21

Hence the standard deviation is:

S = \sqrt{\frac{(17-21)^2 + (26-21)^2 + (28-21)^2 + \cdots + (21-21)^2 + (23-21)^2}{9}} = 8.22

More can be learned about the standard deviation of a data-set at brainly.com/question/12180602

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2 years ago
Tell whether each equation has one, zero, or infinitely many solutions.<br>5(x - 3) +6= 5x - 9​
yuradex [85]

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3 years ago
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salantis [7]

Answer:

1. 424

2.471

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Step-by-step explanation:

1. 10×12+8×10+8×13+8×15

=120+80+104+120

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2. 5×24+9×24+9×15+9×15

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3. 24×16+2×24+2×20+2×20

=384+48+40+40

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3 years ago
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