Question

AP Study: The rate of change R, in km/hr, of the altitude of a hot air balloon is given by R(t)=t^3−4t^2+6,0≤t≤4, where t measures in hours. Assume the hot air balloon is initially at ground level. a. What is the maximum altitude of the balloon during the interval [0,4] ?

Answer 1

To find the maximum altitude of the balloon, we need to find the maximum value of the rate of change function. We can do this by taking the derivative of the function and setting it equal to 0. Substituting this value back into the original function gives us the maximum altitude.

Step-by-step explanation:

To find the maximum altitude of the hot air balloon, we need to find the maximum value of the rate of change function during the interval [0,4]. The rate of change function is given by:

R(t) = t^3 - 4t^2 + 6

To find the maximum value, we can take the derivative of the function and set it equal to 0. The derivative of the function is:

R'(t) = 3t^2 - 8t

Setting R'(t) = 0 and solving for t gives us t = 8/3.

Substituting this value back into the original function gives us the maximum altitude:

R(8/3) = (8/3)^3 - 4(8/3)^2 + 6

Simplifying this expression gives us the maximum altitude of the balloon during the interval [0,4].