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# Find the rate of change in volume of a right circular cylinder

Given a right circular cylinder whose radius increases at a constant rate and whose altitude is a linear function of the radius. Also, given that the altitude is increasing at a rate three times that of the radius. The volume is increasing at a rate of 1 cubic feet per second when the radius is 6 feet. When the radius is 36 feet the volume is increasing at a rate of cubic feet per second. Find the value of the integer .

The following diagram illustrates the setup: Since the altitude (which we denote ) is a linear function of the radius and increases three times as quickly, we have When , we are given . Thus, we solve for and get .
When , we have So, since implies and we have, Solving for we obtain, Then, when we have 1. M.d.S says:

in a similar way one might prefer to solve by:
dV/dt = dV/dr * dr/dt

you find that h= 3r+3
and you know dr/dt to be a constant (let’s call it k)

when dV/dt =1 , r =6
when dV/dt =n , r=36

so you don’t have to solve for k since you have a system in which you can divide one equation by the other:

n=3*pi*(11880)*k
1=3*pi*(360)*k

dividing one equation by the other, the 3pi and k factors cancel out, and you have
n/1=11880/360
n=33

2. Hteica says:

I mistakenly mistook the problem as calculating a cone, since how could a right circular cone has different radii at different heights? xD

• Artem says:

Lol, same here… I even arrived at the answer – but it was different :D