How quickly is the mass of the tree increasing?

Question

A tree trunk is approximated by a circular cylinder of a height of 50 meters and a diameter of 4 meters. The tree is growing taller at a rate of 1 meter per year and the diameter is increasing at a rate of 2 cm per year. The density of the wood is 1000 Kg per cubic meter.

Answer 1

You want to know how the mass changes with time. You need an equation that connects mass with dimensions:

M = ρV = ρ(πr²h) = ρπd²h/4 since we are given information about d. ρ is density.

dM/dt = (πρ/4)(d²dh/dt + h(2d)dd/dt)

You can now plug in all the given values that are on the right-hand side of the equation. (Answer in kg/yr)

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