Activity 4.4.4.
During a 40-minute workout, a person riding an exercise machine burns calories at a rate of \(c\) calories per minute, where the function \(y = c(t)\) is given by the following information. On the interval \(0 \le t \le 10\text{,}\) the formula for \(c\) is \(c(t) = -0.05t^2 + t + 10\text{;}\) on the interval \(10 \le t \le 30\text{,}\) \(c(t) = 15\text{;}\) and on \(30 \le t \le 40\text{,}\) its formula is \(c(t) = -0.05t^2 + 3t - 30\text{.}\)
(a)
What is the exact total number of calories the person burns during the first 10 minutes of her workout?
(b)
Let \(C(t)\) be an antiderivative of \(c(t)\text{.}\) What is the meaning of \(C(40) - C(0)\) in the context of the person exercising? Include units on your answer.
(c)
Determine the exact average rate at which the person burned calories during the 40-minute workout. Sketch a representation of the average rate at which calories were burned on the provided graph of \(c(t)\text{.}\)
(d)
At what time(s), if any, is the instantaneous rate at which the person is burning calories equal to the average rate at which she burns calories, on the time interval \(0 \le t \le 40\text{?}\)
