Oil spilled from a ruptured tanker spreads in a circle. (Three separate, yet rel

Question

Oil spilled from a ruptured tanker spreads in a circle. (Three separate, yet related, problems.) If the area of the circle increases at a constant rate of 4 miles squared per hour, how fast is the radius of the spill increasing when the area is 15 miles squared? If the radius is increasing at a rate of approximately .2913 miles per hour, how fast is the area of the circle increasing when the area is 15 miles squared? How large is the circle (in terms of area) if the area of the circle is increasing at 4 miles squared per hour and simultaneously the radius is increasing at approximately .2913 miles per hour?

Explanation / Answer

i)
A = pi*r^2
15 = pi*r^2
r = 2.185 mile

A = pi*r^2
differentiate with respect to t,
dA/dt = pi*d/dt (r^2)
dA/dt = 2*pi*r* dr/dt
4 mile^2/hr = 2*pi*2.185*dr/dt
dr/dt = 0.291 mile/hr
Answer: 0.2913 mile/hr

ii)
dA/dt = 2*pi*r* dr/dt
= 2*pi*2.185*0.2913
= 4 mile^2/hour
Answer: 4 mile^2/hour

iii)
dA/dt = 2*pi*r* dr/dt
4 = 2*pi*r*0.2913
r = 2.185mil
A = Pi*r^2
= pi*2.185^2
= 15 mile^2
Answer: 15 mile^2

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