For the sake of this problem, assume that the rate of growth of a beanstalk is proportional to the square root of its current height.

Question

If we let h be the current height (in feet) of the beanstalk and dh/dt is the rate of growth, then according to the assumption

dh/dt ∝ √h

which implies that there exists a constant k called the proportionality constant such that

dh/dt = k√h

Find the particular solution to the given differential equation if initially, the beanstalk is 100

feet tall and it grows to 400 feet after 5 days. Here, t is time in days. You need to find the

value of k using the second initial condition stated in the problem.

Answer 1

dh/dt = k√h;

dh/√h = kdt; ∫h^-1/2dh = ∫kdt; 2h^1/2 = kt + C.

Now we have: at t = 0 h = 100; so, 2√100 = C, C = 20

2√h = kt + 20.

Then if t= 5 h = 400 and 2√400 = 5k + 20; 5k = 20; k = 4

2√h = 4t + 20; √h = 2t + 10; h = (2t + 10)².