Question

A tree trunk is approximated by a circular cylinder of height 100100 meters and diameter 33 meters. The tree is growing taller at a rate of 11 meters per year and the diameter is increasing at a rate of 55 cm per year. The density of the wood is 50005000 kg per cubic meter.

how quickly is the mass of the tree increasing?

Answer 1

I'm assuming that height of cylinder is 100 m and not 100100 m (Similarly diameter is 3 m and not 33 meter), If this is not the case than let me know through comments I will update.

We know that mass of tree will be:

Mass = density*Volume

Given density of tree = p = 5000 kg/m^3

Volume of cylindrical tree trunk = pi*r^2*h, So

M = p*pi*r^2*h

D = diameter of trunk = 2*r, So

M = (p*pi/4)*D^2*h

Now given that height of cylinder is increasing at a rate of, dh/dt = 1 m/year

diameter of tree is increasing at a rate of, dD/dt = 5 cm/year = 0.05 m/year

Now rate at which mass of tree is increasing will be:

M = (p*pi/4)*D^2*h

dM/dt = (p*pi/4)*2*D*h*(dD/dt) + (p*pi/4)*D^2*(dh/dt)

dM/dt = (p*pi/4)[2*D*h*(dD/dt) + D^2*(dh/dt))]

Given that at a particular time height of tree = 100 m & Diameter of tree = 3 m, So at that time

dM/dt = (5000*pi/4)[2*3*100*0.05 + 3^2*1]

dM/dt = 153152.64 kg/year

Let me know if you've any query.