Rate of decreasing shadow problem solved with derivative and similar triangles

Spherical Balloon Solution
Known values given: r = 3 $\frac{dV}{dt}$ = 36π in³/minute Use volume of a sphere formula $\frac{4}{3}$ π r³ = V Take derivative. Solve for $\frac{dr}{dt}$ $\frac{4}{3}$ π 3r² $\frac{dr}{dt}$ = $\frac{dV}{dt}$ 4π r² $\frac{dr}{dt}$ = $\frac{dV}{dt}$ Substitute: r = 3 $\frac{dV}{dt}$ = 36π 4π * 3² * $\frac{dr}{dt}$ = 36π 36π $\frac{dr}{dt}$ = 36π $\frac{dr}{dt}$ = 1 The radius of the balloon is growing at a rate of 1 inch per minute.
159R

Spherical Balloon
Solution

Known values given:
r = 3 $\frac{dV}{dt}$ = 36π in³/minute
Drawing of a balloon with r radius indicated.

Use volume of a sphere formula
$\frac{4}{3}$ π r³ = V

Take derivative. Solve for $\frac{dr}{dt}$
$\frac{4}{3}$ π 3r² $\frac{dr}{dt}$ = $\frac{dV}{dt}$
4π r² $\frac{dr}{dt}$ = $\frac{dV}{dt}$

Substitute: r = 3 $\frac{dV}{dt}$ = 36π
4π * 3² * $\frac{dr}{dt}$ = 36π
36π $\frac{dr}{dt}$ = 36π
$\frac{dr}{dt}$ = 1

The radius of the balloon is growing
at a rate of 1 inch per minute.

159R

Spherical Balloon Solution

Spherical Balloon
Solution