volume of wood in trees

Question

The volume of wood V in a tree varies jointly as the height h and the square of the girth g. (girth is the distance around the tree.) If the volume of a redwood tree is 216 m^3 when the height is 30 m and the girth is 1.5 m, what is the height of a tree whose volume is 960m^3 and girth is 2 m?

Answer 1

From reading the first line we can come up with an equation.

 

V ~ h*g²

 

to make this into an equation we add a constant of proportionality (k).

 

V = k*h*g²

 

Now use the values in the 2nd line to determine what k is.

 

k = V/(h*g²) = 216/(30*1.5²)=3.2.

 

note all the values were given in meters, so no need to change any of the units.

 

now we have an equation

 

V = 3.2*h*g², which can be used to find the height of the tree in the 3rd line.

 

solving for h,   h=V/(3.2*g²) and plugging in the values

 

h=960/(3.2*2²)=75   again all units are in meters..  Hence the height is 75 meters.

Answer 2

V is proprotional to h and g^2

V= K x h x g^2     k is constant of proportionality

Given data

216 = K x 30 x 1.5^2

Solve for K

K= 3.2

Use K to find unknown data

960= 3.2 x h x 2^2

h = 75m

:)