The height h, in feet, of a certain tree increases at a rate that is inversely proportional to time t, in years. The height of the tree is 8 feet at the end of 3 years, and reaches 23 feet at the end of 6 years. Write a differential equation describing the rate of change of the height of the tree, where k is the constant of proportionality (to be determined).
A) feet per year
B) feet per year
C) feet per year
D) feet per year
E) feet per year

Question

The height h, in feet, of a certain tree increases at a rate that is inversely proportional to time t, in years. The height of the tree is 8 feet at the end of 3 years, and reaches 23 feet at the end of 6 years. Write a differential equation describing the rate of change of the height of the tree, where k is the constant of proportionality (to be determined).
A) feet per year 
B) feet per year 
C) feet per year 
D) feet per year 
E) feet per year

Quizplus · Accepted Answer

Let's assume h is the height of the tree at time t. Then, the rate of change of the height of the tree, dh/dt, is inversely proportional to time t:
dh/dt = k/t
where k is the constant of proportionality.

Using the information provided, we can set up a system of equations:
h(3) = 8
h(6) = 23

We can solve for k by using the first equation:
8 = k/3
k = 24

Substituting k into our differential equation, we get:
dh/dt = 24/t

Therefore, the rate of change of the height of the tree is 24/t feet per year. The closest option is D, 11⅓ feet per year, which can be approximated by plugging in t = 2.5 (the average of 3 and 6):
24/2.5 ≈ 9.6 feet per year.

The Height H, in Feet, of a Certain Tree Increases