How quickly is the mass of the tree increasing?

Question

A tree trunk is approximated by a circular cylinder of height 50 meters and diameter 3 meters. The tree is growing taller at a rate of 4 meters per year and the diameter is increasing at a rate of 1 cm per year. The density of the wood is 4000 Kg per cubic meter.

How quickly is the mass of the tree increasing?

EDIT: I've tried 156000pi but I'm still getting marked incorrectly :(

Answer 1

Now let d = density = M/V => M = d * V

 

Now V = πr² * h => M = π * d * r² * h

 

The related rate for the mass is given by dM/dt = π * d * (2r * h * dr/dt +r² *dh/dt)

 

Given r = 1.5 m, h = 50 m, dr/dt = 0.005 m/yr, dh/dt = 4 m/yr and d = 4000 Kg/m³

 

dM/dt = 4000π * (2 * 1.5 * 50 * .005 + 1.5² * 4) = 4000π * 9.75 = 39000π Kg/yr = 122,522.1 Kg/yr

Answer 2

When I worked this out for you earlier, I missed the part where it says the "diameter" is increasing at 0.01 m/yr.  We want the radius, not the diameter.  The radius is increasing at half the rate of the diameter: dr/dt = 0.005 m/yr.  Plug in 0.005 vice 0.01 and you get dm/dt = 4000*pi*37.5 (instead of 39) = 150,000pi = 471,238.9 kg/yr.