A research drone ascends vertically so that its altitude (in meters) is a…

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

A research drone ascends vertically so that its altitude

h

(in meters) is a differentiable function of time

t

(in seconds). The ambient temperature

T

(in degrees Celsius) at the drone’s location is a differentiable function of altitude

h

. A scientist records the following information at

t=5\text{ s}

: • The drone is

600\text{ m}

above ground, so

h(5)=600

. • The drone’s vertical velocity is

\dfrac{dh}{dt}(5)=12\text{ m/s}

. • Atmospheric data indicate that near

h=600\text{ m}

, temperature decreases with altitude at a rate of

0.006\text{ }^{\circ}\!\text{C per meter}

. Let

T(t)=T(h(t))

. (a) Explain the physical significance of

\dfrac{dT}{dh}

at an altitude of

600\text{ m}

, including its sign. (b) Determine

\dfrac{dT}{dt}

t=5\text{ s}

. Show all steps clearly and include proper units in your final answer. (c) Interpret the meaning of your result from part (b) in the context of the drone’s flight. (d) Given that

\dfrac{d^{2}h}{dt^{2}}(5)=-3\text{ m/s}^2

and that

\dfrac{d}{dh}\left(\dfrac{dT}{dh}\right)

is negligible near

h=600\text{ m}

, discuss whether

\dfrac{d^{2}T}{dt^{2}}(5)

is likely to be positive, negative, or approximately zero. Provide a reasoned argument without performing a full calculation.

Related theory

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