Question

a 5 - foot - tall woman walks at 8 ft/s toward a street light that is 30 ft above the ground. what is the rate of change of the length of her shadow when she is 18 ft from the street light? at what rate is the tip of her shadow moving?
the rate of change of the length of the womans shadow is when she is 18 ft from the street light.
(type an exact answer in simplified form.)

Explanation:

Step1: Set up the proportion using similar triangles

Let \(x\) be the distance of the woman from the street - light and \(y\) be the length of her shadow.
Since the triangles are similar, \(\frac{5}{30}=\frac{y}{x + y}\).
Cross - multiply: \(5(x + y)=30y\).
Expand: \(5x+5y = 30y\).
Rearrange: \(5x=25y\), so \(y=\frac{1}{5}x\).

Step2: Differentiate with respect to time \(t\)

Differentiate \(y=\frac{1}{5}x\) with respect to \(t\).
Using the chain rule \(\frac{dy}{dt}=\frac{1}{5}\frac{dx}{dt}\).
We know that \(\frac{dx}{dt}=- 8\) (negative because \(x\) is decreasing as she walks towards the light).

Answer:

\(\frac{dy}{dt}=\frac{1}{5}\times(-8)=-\frac{8}{5}\text{ ft/s}\)