The area of the circular cross-section of the clay is decreasing at a rate of 72π square inches per minute.
The area of the circular cross-section of the clay can be found using the formula A = πr2. To find the rate at which the area is decreasing, we need to take the derivative of the area with respect to time. Given that the radius is decreasing at a rate of 12 inches per minute, we can substitute that value into the derivative formula. Let's calculate:
At time t=c, the radius r = 3 inches and the height h = 8 inches.
The formula for the area of the circular cross-section is A = πr2.
Take the derivative of the area with respect to time, dA/dt:
dA/dt = 2πr(dr/dt)
Substitute r = 3 inches and dr/dt = -12 inches/minute (since the radius is decreasing):
dA/dt = 2π(3)(-12) = -72π square inches per minute
Therefore, at time t=c, the area of the circular cross-section of the clay is decreasing at a rate of 72π square inches per minute.
--The given question is incomplete, the complete question is
"A sculptor uses a constant volume of modeling clay to form a cylinder with a large height and a relatively small radius. The clay is molded in such a way that the height of the clay increases as the radius decreases, but it retains its cylindrical shape. At time t=c, the height of the clay is 8 inches, the radius of the clay is 3 inches, and the radius of the clay is decreasing at a rate of 12 inch per minute. (a) at time t=c, at what rate is the area of the circular cross section of the clay decreasing with respect to time? show the computations that lead to your answer. Indicate units of measure."--
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