The height (in meters) of a research balloon above sea-level during the first five…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
The height h (in meters) of a research balloon above sea-level during the first five minutes of ascent is modeled for 0≤t≤300 seconds by the differentiable function
h(t)=120te−0.004t+50sin(150πt)+85.
Assume the model accurately describes the balloon’s motion only on this closed time interval.
(a) Calculate h(0) and h(300), and interpret these values with respect to the balloon’s flight.
(b) Find all values of t in (0,300) at which the balloon could reach a local maximum or minimum height. Support your work analytically.
(c) Determine the absolute maximum height the balloon attains during the first five minutes and the time at which this occurs. Provide justification that your answer represents the greatest height on the entire interval.
(d) Explain, without additional calculation, why it is sufficient to examine only the points identified in parts (a) and (b) when seeking the absolute maximum height on the closed interval.