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Debora [2.8K]
3 years ago
9

PLS HELP ME ASAP WITH 2!! (MUST SHOW WORK!!) + LOTS OF POINTS!!

Mathematics
1 answer:
pickupchik [31]3 years ago
6 0
The answer is b, n less than 16
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A sphere of radius r is cut by a plane h units above the equator, where
Anika [276]
Consider the top half of a sphere centered at the origin with radius r, which can be described by the equation

z=\sqrt{r^2-x^2-y^2}

and consider a plane

z=h

with 0. Call the region between the two surfaces R. The volume of R is given by the triple integral

\displaystyle\iiint_R\mathrm dV=\int_{-\sqrt{r^2-h^2}}^{\sqrt{r^2-h^2}}\int_{-\sqrt{r^2-h^2-x^2}}^{\sqrt{r^2-h^2-x^2}}\int_h^{\sqrt{r^2-x^2-y^2}}\mathrm dz\,\mathrm dy\,\mathrm dx

Converting to polar coordinates will help make this computation easier. Set

\begin{cases}x=\rho\cos\theta\sin\varphi\\y=\rho\sin\theta\sin\varphi\\z=\rho\cos\var\phi\end{cases}\implies\mathrm dx\,\mathrm dy\,\mathrm dz=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi

Now, the volume can be computed with the integral

\displaystyle\iiint_R\mathrm dV=\int_0^{2\pi}\int_0^{\arctan\frac{\sqrt{r^2-h^2}}h}\int_{h\sec\varphi}^r\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\varphi\,\mathrm d\theta

You should get

\dfrac{2\pi}3\left(r^3\arctan\dfrac{\sqrt{r^2-h^2}}h-\dfrac{h^3}2\left(\dfrac{r\sqrt{r^2-h^2}}{h^2}+\ln\dfrac{r+\sqrt{r^2-h^2}}h\right)\right)
5 0
3 years ago
What is the solution for 4/5x-2=x-1/3
Veseljchak [2.6K]

Answer: Is -25/3 or -8.3 repeating.

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3 years ago
Which of the following best describes the expression 12(X+3)
nydimaria [60]

Answer: 12x+36

Step-by-step explanation:

If you use distributive property, 12 x X is 12x and 12 x 3 is 36 so you get 12x+36

5 0
3 years ago
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Hello please help and i give you brainliest
Hunter-Best [27]

Answer:

16

Step-by-step explanation:

x+6

In this step instead of puting x you relace it with 10 which is what x equals

10+6

16

8 0
3 years ago
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What is the answer to the question 3x+5x
Elodia [21]

Step-by-step explanation:

3x+5x

8x

3x and 5x are same term because they have same bases so they can be added as above .

7 0
3 years ago
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