Calculus Question

Question

(a) If A is the area of a circle with radius r and the circle expands as time passes, find dA/dt

 in terms of dr/dt.

(b) Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 1 m/s, how fast is the area of the spill increasing when the radius is 37 m?

 

Answer 1

Hi Sonam, the area A(r) of a circle of radius r is given by: A(r) = π*r² Thus, using the chain rule, dA/dt = dA/dr * dr/dt = 2πr*dr/dt. The answer to part (b) falls out when you substitute r = 37m and dr/dt = 1 m/s.

Note that the formula for dA/dt includes the product of circumference and the differential width dr: 2πr*dr. This makes sense geometrically as this product represents the differential area during expansion.

Robert