I have number 9 , i just need number 10

Use cylindrical coordinates. find the volume of the solid that is enclosed by the cone z = x2 + y2 and the sphere x2 + y2 + z2 =

sashaice [31]

Let $R$ be the solid. Then the volume is

$\displaystyle\iiint_R\mathrm dV=\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=\sqrt8}\int_{\zeta=r^2}^{\zeta=\sqrt{72-r^2}}r\,\mathrm d\zeta\,\mathrm dr\,\mathrm d\theta$

which follows from the facts that

$\begin{cases}x=r\cos\theta\\y=r\sin\theta\\z=\zeta\end{cases}\implies\mathrm dx\,\mathrm dy\,\mathrm dz=r\,\mathrm dr\,\mathrm d\theta\,\mathrm d\zeta$
(by computing the Jacobian)

and

$z=x^2+y^2=r^2\implies z+z^2=72\implies z=-9\text{ or }z=8$
(we take the positive solution, since it's clear that $R$ lies above the $x$ - $y$ plane)
$r^2+z^2=72\implies z=\pm\sqrt{72-r^2}$
(again, taking the positive root for the same reason)
$z=r^2\implies 8=r^2\implies r=\sqrt8$

$\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=\sqrt8}\int_{\zeta=r^2}^{\zeta=\sqrt{72-r^2}}r\,\mathrm d\zeta\,\mathrm dr\,\mathrm d\theta$
$=\displaystyle2\pi\int_{r=0}^{r=\sqrt8}\int_{\zeta=r^2}^{\zeta=\sqrt{72-r^2}}r\,\mathrm d\zeta\,\mathrm dr$
$=\displaystyle2\pi\int_{r=0}^{r=\sqrt8}r(\sqrt{72-r^2}-r^2)\,\mathrm dr$
$=\displaystyle2\pi\int_{r=0}^{r=\sqrt8}(r\sqrt{72-r^2}-r^3)\,\mathrm dr$
$=\dfrac{32(27\sqrt2-35)\pi}3$

7 0

3 years ago

Dan is 4 years older than his brother Gabriel. Together their ages equal 22. How old is each boy?

zmey [24]

(4+x)+x=22
= 4+2x=22
= 2x = 18
= x = 9.

Gabriel is 9 years old, and Dan is 4 + 9 or 13 years old.

3 0

3 years ago

Read 2 more answers

15. The average speed that a tsunami (a large tidal wave) travels is represented

irina [24]

Correct question :

The average speed that a tsunami (a large tidal wave) travels is represented

by the function $s = (200d)^1^/^2$ where s is the speed (in miles per hour) that the tsunami is traveling and d is the average depth (in feet) of the wave.

a. Find the inverse of the function.

b. Find the average depth of the tsunami when the recorded speed of the wave is 250 miles per hour.

Answer:

a) $d = \frac{s^2}{200}$

b) 312.5 ft

Step-by-step explanation:

Given:

Average speeds, $s = (200d)^1^/^2$

a) To find the inverse of the function.

$s = (200d)^1^/^2$

s² = 200d

200d = s²

$d = \frac{s^2}{200}$

Therefore, inverse of the function =

$d = \frac{s^2}{200}$

b) average depth when speed is 250 miles per hour.

Average depth = d

Therefore, let's use the formula :

$d = \frac{s^2}{200}$

$= \frac{250^2}{200}$

$= \frac{62500}{200}$

d = 312.5 feet

The average depth when speed is 250 miles per hour is 312.5 ft

3 0

3 years ago

Read 2 more answers

PLZ PLZ PLZ HELP ME GET THIS RIGHT ILL GIVE 40 POINTS AND BRAINLY IF THERE BOTH RIGHT, I REALLY NEED HELP I SUCK AT GEOMETRY!!!

tia_tia [17]

I believe they are both 120

8 0

2 years ago

Write three DIFFERENT properties that are equivalent to 3 4

stich3 [128]

Answer:

6/8

9/12

12/16

Step-by-step explanation:

6 0

3 years ago