Question

The spread of oil leaking from a tanker after t hours is in the shape of a circle with radius
r(t) = 200 square root t+3, - 600 (in feet).
A. Find the time it takes for the leak to spread at a radius of 1800ft.
B. Find the area A of the oil leak as a function of time t.

Answer 1

A.
substitute r = 1800 to the equation:
1800 = 200*sqrt(t+3) - 600
2400 = 200*sqrt(t+3)
12 = sqrt(t+3)
144 = t+3
t = 141 hours

B.
recall that area of circle is just A = pi*r^2. thus we square both sides and multiply by pi:
r(t) = 200*sqrt(t+3) - 600
r^2 = 40000(t+3) - 240000*sqrt(t+3) - 360000
pi*r^2 = A = pi*[40000(t+3) - 240000*sqrt(t+3) - 360000]
we can further simplify this as,
A = pi*[40000t + 120000 - 240000*sqrt(t+3) - 360000]
A = pi*[40000t - 240000*sqrt(t+3) - 240000)

hope this helps~ :)