The radius r of a circular disc is increasing at the rate of 0.5cm/sec. At what rate is the area of the disc increasing when its radius is 6cm?

Answer Details

We are given that the radius of a circular disc is increasing at a rate of 0.5cm/sec. We need to find the rate at which the area of the disc is increasing when the radius is 6cm. The formula for the area of a circle is A = πr^2, where A is the area and r is the radius. We can use the chain rule of differentiation to find the rate of change of the area with respect to time. dA/dt = dA/dr * dr/dt We know that dr/dt = 0.5 cm/sec (given) and we need to find dA/dt when r = 6 cm. dA/dr = 2πr (differentiating A = πr^2 with respect to r) So, dA/dt = (2πr) * (0.5) = πr * 1 = 6π cm^2/sec (substituting r = 6cm) Therefore, the rate at which the area of the disc is increasing when its radius is 6cm is 6π cm^2/sec. is the correct answer.