- #1

AN630078

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- Homework Statement
- Hello, I have been practising some related rates problems and have found the following problem which I am a little wary of. I feel a little uncertain of my solution and would greatly appreciate any advice.

The radius of a sphere is increasing at a constant rate of 3cms^-1. Given that the radius of the sphere is 5cm find in terms of π the rates at which its surface area and volume are increasing.

- Relevant Equations
- Volume of sphere= 4/3πr^3

Surface area of sphere = 4πr^2

The surface area of the sphere is 4πr^2.

dr/dt is given as 3cm^-1.

dS/dt=dS/dr*dr/dt

Differentiating 4πr^2 is dS/dr= 8πr

dS/dt=8πr*3

dS/dt=24πr

Given that r=5 dS/dt=24π*5=120 π

The volume of the sphere is 4/3πr^3, differentiating which is dV/dr=4πr^2

dV/dt=dV/dr*dr/dt

dV/dt= 4πr^2*3

dV/dt=12πr^2

Given that r=5, dV/dt=12π*(5^2)=300π

I honestly do not know whether this is correct and have only been introduced to solving similar problems through reading their mark schemes which is why I am very unsure of my method. I found a similar problem, here at https://www.toppr.com/ask/question/the-volume-of-a-sphere-is-increasing-at-the-rate-of-3-cubic-centimetre-per/

but following this approach of finding the rate at which the volume is increasing first and using this to find dr/dt I got a different answer and have now confused myself.

Just to check, would dr/dt=3cm^-1? I think this value is what is confusing me most of all. I would very much appreciate if anyone could tidy up my workings or suggest an alternative method with greater clarity

dr/dt is given as 3cm^-1.

dS/dt=dS/dr*dr/dt

Differentiating 4πr^2 is dS/dr= 8πr

dS/dt=8πr*3

dS/dt=24πr

Given that r=5 dS/dt=24π*5=120 π

The volume of the sphere is 4/3πr^3, differentiating which is dV/dr=4πr^2

dV/dt=dV/dr*dr/dt

dV/dt= 4πr^2*3

dV/dt=12πr^2

Given that r=5, dV/dt=12π*(5^2)=300π

I honestly do not know whether this is correct and have only been introduced to solving similar problems through reading their mark schemes which is why I am very unsure of my method. I found a similar problem, here at https://www.toppr.com/ask/question/the-volume-of-a-sphere-is-increasing-at-the-rate-of-3-cubic-centimetre-per/

but following this approach of finding the rate at which the volume is increasing first and using this to find dr/dt I got a different answer and have now confused myself.

Just to check, would dr/dt=3cm^-1? I think this value is what is confusing me most of all. I would very much appreciate if anyone could tidy up my workings or suggest an alternative method with greater clarity