Question

A lumber company needs to estimate the volume of wood a load of timber will produce. The supervisor knows that the volume of wood in a tree varies jointly as the height h and the square of the tree’s girth g. The supervisor observes that a tree 40 meters tall with a girth of 1.5 meters produces 288 cubic meters of wood.
a. Write an equation that represents this situation.
b. What volume of wood can the supervisor expect to obtain from 50 trees averaging 75 meters in height and 2 meters in girth?

Answer 1

Answer with Step-by-step explanation:

Let the volume be represented by 'V'

Since it is given that Volume varies with height 'h' and square of the girth 'g' we can write

$V=khg^2$ where 'k' is a constant of proportionality

Now it is given that volume is 288 cubic meters for  h= 40 meters and g =1.5 meters we can write

$288=k\times 40\times 1.5^2\\\\\therefore k=\frac{288}{40\times 1.5^{2}}=3.2$

Thus the volume of the timber as a relation is

$V=3.2\times hg^{2}$

Part b)

Volume of 50 trees with height 75 meters and girth 2 meters is

$V=50\times 3.2\times 75\times 2^{2}\\\\\therefore V=48000m^{3}$