1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Vlad [161]
3 years ago
12

A sphere of radius r is cut by a plane h units above the equator, where

Mathematics
1 answer:
Anika [276]3 years ago
5 0
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)
You might be interested in
The diameter of your bicycle wheel is 34 inches. How far will you move in one turn of your wheel? What is the distance covered i
irga5000 [103]
In one turn of the wheel, you will move the circumference of the wheel. To find the circumference, you just do π × diameter, so π × 34 = 106.8 inches. In 5 turns, you will just go 5 times the circumference, so you multiply by 5 to get 534.1 inches. I hope this helps!
6 0
3 years ago
5 An army cook wants to make pudding
photoshop1234 [79]

Answer:

27

Step-by-step explanation:

First you dvide 864 by 4  to get 216. Next you multiply the 216 by 2 to find that there are 432 cups in 864 servings. Finally you divide by 16 to turn 432 cups into 27 gallons.

I hope this helps!

3 0
3 years ago
Read 2 more answers
Select the expression that is equivalent to x + x + y × y × y. Question 11 options: (a)x^2 + 3y (b)2x + 3y (c)x^2 + y^3 (d)2x +
beks73 [17]

x+x+y.y.y = 2x+y^3 ( y cube)

So, The answer is (d)

7 0
3 years ago
What is 5x+4=26 I don’t understand it please help thanks
babunello [35]

Answer: 4.4

Step-by-step explanation:

First you need to take 4 from both sides and then your left with 5x=22

Then you divide from both sides and you get 4.4.

That is your final answer.

3 0
3 years ago
Read 2 more answers
Juanita has a storage closest at her shop with extra bottles of lotion and shower gel. Some are scented and some are unscented.
Eddi Din [679]
<span>P(shower gel | scented) = 42%</span>
7 0
3 years ago
Read 2 more answers
Other questions:
  • Use synthetic division to find P(3) for P(x) =
    14·1 answer
  • Dennis wants to buy a card for his wife Dennis calculates the amount of the card has $4.50 the actual price of card is $4 what i
    8·1 answer
  • Find the slope of (1,1) and (3,1)
    8·1 answer
  • I need help and i need it fast!
    5·1 answer
  • Which expressions are equal to 8 × 10to the 5 power? Select all that apply. 8 × 1,000 8 × 10,000 8 × 100,000 8 × 10 × 10 × 10 ×
    5·1 answer
  • 165. How many moles of sodium will react with water to
    7·2 answers
  • Please help asap! will give brainliest
    7·1 answer
  • Slope intercept form
    7·1 answer
  • 5(x+6)-8x=3(10-x)n no solution, one solution, or all real numbers are solutions
    14·1 answer
  • C is the circumcenter of APQR.<br> PC=3x+7, RC=5x-15, &amp; QC=51-X<br> x=[?]
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!