Question

A 7 ft tall person is walking away from a 20 ft tall lamppost at a rate of 5 ft/sec. Assume the scenario can be modeled with right triangles. At what rate is the length of the person's shadow changing when the person is 16 ft from the lamppost?In similar triangles, both the two triangles must satisfy the two properties. One is the side proportional, and the other is equal in angles. There are three criteria in similarity. They are AA similarity, SSS similarity, and SAS similarity. The below one satisfies the AA similarity.

Answer 1

The change in rate of length of shadow is 2.692 ft/sec.

Given,

Height of tall person =7 ft

Height of tall lamp post = 20 ft

Rate at which tall person walks = 5 ft/sec

Let,

Distance between tall person and lamp post be x ft

the length of shadow be y ft

From similar triangles,

$\frac{x+y}{20}=\frac{y}{7}\\\\7(x+y)=20y\\\\7x+7y=20y\\\\13y=7x\\\\y=\frac{7x}{13}$

Differentiating on both sides with respect to time 't'

$\frac{dy}{dt}=\frac{7}{13}\frac{dx}{dt}$

here, $\frac{dx}{dt}$ is nothing but the change in distance between tall person and lamppost it means rate at which tall person walks=5 ft/sec

$\frac{dy}{dt}=\frac{7}{13}*5\\\\\frac{dy}{dt}=\frac{35}{13}=2.692\ ft/sec$

Thus, the change in rate of length of shadow is 2.692 ft/sec.

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