Question #4613

Hello,
When the dr/dt for sphere is constant and r=at + b, how does dV/dt change over time?
This is my work so far:
dr/dt = a
V=(4/3)(pi)r^3
dV/dt = 4(pi)r^2 (dr/dt)
dV/dt = 4(pi)(at + b)^2 (a)
Well, if radius is steadily decreasing, is change in volume decreasing constantly.
I know in my work, since there is a variable t, volume would not change at a constant rate. But am I thinking about it correctly?

Expert's answer

Let's assume that a and b are greater than zero.

As you noted, dV/dt = 4a(pi)(at + b)^2. It's greater than zero anyway. So, V would increase while increasing t. Let's get d(dV/dt)/dt:

d(dV/dt)/dt = 8(a^2)(pi)(at+b). It's also greater then zero, so, dV/dt would increase while increasing t, similarly.

As you noted, dV/dt = 4a(pi)(at + b)^2. It's greater than zero anyway. So, V would increase while increasing t. Let's get d(dV/dt)/dt:

d(dV/dt)/dt = 8(a^2)(pi)(at+b). It's also greater then zero, so, dV/dt would increase while increasing t, similarly.

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