A straight ladder 25 ft long rests against a vertical wall on a smooth horizontal ground

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

Problem

A straight ladder 25 ft long rests against a vertical wall on a smooth horizontal ground. The bottom of the ladder is pulled directly away from the wall at a constant rate of 3 ft/s. At the instant when the bottom of the ladder is 7 ft from the wall, the position of the top of the ladder on the wall is 24 ft above the ground. Let

x(t)

denote the distance (in feet) from the bottom of the ladder to the wall at time

t

seconds, and let

y(t)

denote the height (in feet) of the top of the ladder on the wall at time

t

. (a) Write an equation that relates

x

and

y

, and use differentiation with respect to time to obtain a relationship between

\dfrac{dx}{dt}

and

\dfrac{dy}{dt}

. No numerical substitution is required in this part. (b) Determine the instantaneous rate of change of

y

with respect to time,

\dfrac{dy}{dt}

, at the instant when

x=7

ft. (c) Interpret the sign of

\dfrac{dy}{dt}

found in part (b) within the physical context of the ladder. Specifically, state whether the top of the ladder is moving upward or downward at that instant and justify your conclusion. (d) Compute the instantaneous rate of change of the area of the right triangle formed by the ladder, the wall, and the ground at the same instant. State appropriate units for your answer.

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