Question

Sketch the situation if necessary and use related rates to solve. A 6 ft tall person walks away from a 10 ft lamppost at a constant rate of 5 ft/s. What is the rate (in ft/s) that the tip of the shadow moves away from the pole when the person is 15 ft away from the pole

Answer 1

Answer:

12.5 ft/s

Step-by-step explanation:

$\dfrac{dx}{dt}=\text{Velocity of of person}=5\ \text{ft/s}$

As the two triangles in the figure are similar to each other we havec

$\dfrac{10}{y}=\dfrac{6}{y-x}\\\Rightarrow 10(y-x)=6y\\\Rightarrow 10y-10x=6y\\\Rightarrow -10x=-4y\\\Rightarrow y=\dfrac{10}{4}x$

Differentiating with respect to time we have

$\dfrac{dy}{dt}=\dfrac{10}{4}\dfrac{dx}{dt}\\\Rightarrow \dfrac{dy}{dt}=\dfrac{10}{4}\times 5\\\Rightarrow \dfrac{dy}{dt}=12.5\ \text{ft/s}$

Rate at which the tip of the shadow moves away from the pole is 12.5 ft/s.