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?

Answer 1

Answer:

The rate of change of the shadow length of a person is 2.692 ft/s

Solution:

As per the question:

Height of a person, H = 20 ft

Height of a person, h = 7 ft

Rate = 5 ft/s

Now,

From Fig.1:

b = person's distance from the lamp post

a = shadow length

Also, from the similarity of the triangles, we can write:

$\frac{a + b}{20} = \frac{a}{7}$

$a = \frac{7}{13}b$

Differentiating the above eqn w.r.t t:

$\frac{da}{dt} = \frac{7}{13}.\frac{db}{dt}$

Now, we know that:

Rate = $\frac{db}{dt} = 5\ ft/s$

Thus

$\frac{da}{dt} = \frac{7}{13}.\times 5 = 2.692\ ft/s$