Question

a region is bounded by two concentric circles, as shown by the shaded region in the figure above. the radius of the outer circle, rr, is increasing at a constant rate of 22 inches per second. the radius of the inner circle, rr, is decreasing at a constant rate of 11 inch per second. what is the rate of change, in square inches per second, of the area of the region at the instant when rr is 44 inches and rr is 33 inches?

Answer 1

The rate of change, in square inches per second, of the area of the region at the instant when R is 44 inches and r is 33 inches is

8363 square inches

How to find the rate of change

The rate of change is the derivative, the rate of change is calculated by differentiation the area

formula for area of concentric circle is given by

Area A = π(R^2 - r^2) =

R = radius of the inner circle

r = radius of the outer circle

δA/δt = δA/δRδr * δRδr/δt

δA/δt = π * (2RδR/δt - 2rδr/δt)

δA/δt = π * 2(R * δR/δt - r * δr/δt)

where R = 44 and δR/δt = 22

r = 33 and δr/δt = -11

= 2π(44 * 22 - 33 * -11)

= 2π (968 - -363)

=  2π (1331)

= 2662π

= 8362.9196 square inches

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