Let V be the volume of a sphere with the radius r that is changing with respect to time. If dr/dt is constant, is dV/dt constant?
Can you also explain why it is or isn't? I want to learn so I can have the knowledge for future problems.
Please help and Thank you!
Can you also explain why it is or isn't? I want to learn so I can have the knowledge for future problems.
Please help and Thank you!
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V = (4/3) * pi * r^3
dV/dt = 4 * pi * r^2 * dr/dt
Now, the question is simple. If dr/dt is some constant value (let's call it p) will dV/dt be some other constant value?
dV/dt = 4 * pi * r^2 * p
dV/dt = 4 * p * pi * r^2
The answer is no. Proof. Pick any 2 values for r
r = 1
dV/dt = 4p * pi * 1 = 4p * pi
r = 10
dV/dt = 4p * pi * 100 = 400 * p * pi
Since 4p * pi does not equal 400p * pi, and since both are values for dV/dt, then dV/dt is not constant just because the change in the radius is constant.
dV/dt = 4 * pi * r^2 * dr/dt
Now, the question is simple. If dr/dt is some constant value (let's call it p) will dV/dt be some other constant value?
dV/dt = 4 * pi * r^2 * p
dV/dt = 4 * p * pi * r^2
The answer is no. Proof. Pick any 2 values for r
r = 1
dV/dt = 4p * pi * 1 = 4p * pi
r = 10
dV/dt = 4p * pi * 100 = 400 * p * pi
Since 4p * pi does not equal 400p * pi, and since both are values for dV/dt, then dV/dt is not constant just because the change in the radius is constant.