Related Rates Volume of Spherical Balloon.

Question

The question I was given was:

The volume of a spherical balloon is increasing at a constant rate of $0.78$ inches per minute. At the instant when the radius is $3.20$ inches, the radius is increasing at a rate of

A) $0.006$ in/min

B) $0.019$ in/min

C) $0.419$ in/min

D) $6.273$ in/min

E) $100.37$ in/min

I know that $dV/dt = 0.78, r = 3.2$, and I am trying to find $dr/dt$, but I am confused on how to start it. Do I find the derivative of the volume of a sphere? (Volume of a sphere: $(4/3)(\pi)(r^3)$)

Answer 1

$$V = \frac{4 \pi}{3} r^3$$

so

$$dV = \frac{4 \pi}{3} 3 r^2 dr$$

You know $dV/dt$. Solve for $dr/dt$.

Answer 2

Hint: Think of $V$ and $r$ as being functions of $t$ (time), so that $$V(t) = \frac43\pi \cdot r(t)^3$$ and therefore by the chain rule $$V'(t) = 4\pi\cdot r(t)^2 \cdot r'(t)$$ You know $r(t_0)$ and $V'(t_0)$ at the instant $t_0$ in question, so this will determine $r'(t_0)$ at that instant as well.

Answer 3

Hint: You know that $V = \frac{4}{3}\pi r^{3}$. Use implicit differentiation to take the derivative of both sides with respect to $t$. You should have $\frac{dV}{dt}$ on the left, and something involving $\frac{dr}{dt}$ on the right.