Answer:
The rate of change of the volume
when the height is 9 centimeters and the radius is 6 centimeters is 
Step-by-step explanation:
This is a related rate problem because you know a rate and want to find another rate that is related to it. If 2 variables both vary with respect to time and have a relation between them, we can express the rate of change of one in terms of the other.
From the information given we know:


- The volume of a cone of radius r and height h is given by

We want to find the rate of change of the volume
when the height is 9 centimeters and the radius is 6 centimeters.
Applying implicit differentiation to the formula of the volume of a cone we get
![\frac{dV}{dt}=\frac{1}{3}\pi [r^2\frac{dh}{dt}+2rh\frac{dr}{dt} ]](https://tex.z-dn.net/?f=%5Cfrac%7BdV%7D%7Bdt%7D%3D%5Cfrac%7B1%7D%7B3%7D%5Cpi%20%5Br%5E2%5Cfrac%7Bdh%7D%7Bdt%7D%2B2rh%5Cfrac%7Bdr%7D%7Bdt%7D%20%5D)
Substituting the values we know into the above formula:
![\frac{dV}{dt}=\frac{1}{3}\pi [(6)^2\frac{1}{2}+2(6)(9)\frac{1}{2} ]\\\\\frac{dV}{dt}=\frac{1}{3}\pi[18+54]\\\\\frac{dV}{dt}=\frac{72\pi}{3}=24\pi \:\frac{cm^3}{s}](https://tex.z-dn.net/?f=%5Cfrac%7BdV%7D%7Bdt%7D%3D%5Cfrac%7B1%7D%7B3%7D%5Cpi%20%5B%286%29%5E2%5Cfrac%7B1%7D%7B2%7D%2B2%286%29%289%29%5Cfrac%7B1%7D%7B2%7D%20%5D%5C%5C%5C%5C%5Cfrac%7BdV%7D%7Bdt%7D%3D%5Cfrac%7B1%7D%7B3%7D%5Cpi%5B18%2B54%5D%5C%5C%5C%5C%5Cfrac%7BdV%7D%7Bdt%7D%3D%5Cfrac%7B72%5Cpi%7D%7B3%7D%3D24%5Cpi%20%5C%3A%5Cfrac%7Bcm%5E3%7D%7Bs%7D)
The answer to this question is x = 13/9 = 1.444
Answer:
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Answer:

Step-by-step explanation:
The formula of a volume of a sphere:

R - radius
We have

Substitute and solve for R:
<em>divide both sides by π</em>
<em>multiply both sides by 3</em>
<em>divide both sides by 4</em>
![125=R^3\to R=\sqrt[3]{125}\\\\R=5\ cm](https://tex.z-dn.net/?f=125%3DR%5E3%5Cto%20R%3D%5Csqrt%5B3%5D%7B125%7D%5C%5C%5C%5CR%3D5%5C%20cm)
The formula of a Surface Area os a sphere:

Substitute:

