two similar math problems?

Question

The radius of a sphere is increasing at a rate of 4 mm/s. How fast is the volume increasing when the diameter is 100 mm? Evaluate your answer numerically.
mm^3/s

A street light is mounted at the top of a 15 foot tall pole. A man 6 ft tall walks away from the pole with a speed of 4 ft/s along a straight path. How fast is the tip of his shadow moving when he is 40 ft from the pole?
ft/s

Answer 1

V = 4/3pi r^3
dV/dt= dV/dr*dr/dt = 4pir^2*4 mm^3/s
dV/dt= 126,663.7 mm^3/s

If you call x the distance you can write

x/15 =(x-4*t)/6
6x= 15x-60t so x = 60/9 *t so dx/dt = 60/9 ft/s constant

Answer 2

sorry for my previous answer
volume of sphere= 4/3(pie)r^3
differentiating wrt time
dv/dt=4/3*(pie)*3*r^2*dr/dt

given rate of increase of radius=dr/dt=4mm/s

substituting the values
dv/dt=4/3*(pie)*3*r^2*4
at d= 100 mm ie r= 50 mm
ANS. dv/dt= 125714 mm^3/s

Answer 3

1. V = 4/3pir^3
dv/dt = 4 pi r2 dr/dt we know that dr/dt = 4
dv/dt = 16 pi r^2
when diam = 100, r = 50
dV/dt = 16 pi (50)^2 = 40,000pi mm^3/s