Question

A 5.5-foot-tall woman walks at 4 ft/s toward a street light that is 33 ft above the ground. What is the rate of change of the length of her shadow when she is 12 ft from the street light? At what rate is the tip of her shadow moving?

Answer 1

To find the length of the woman's shadow when she is 12 ft from the street light, we can use similar triangles. The length of her shadow will be 72 ft. The rate of change of the length of the shadow when she is walking towards the street light at 4 ft/s is -24 ft/s.

Step-by-step explanation:

To solve this problem, we can use similar triangles. Let's first find the length of the woman's shadow when she is 12 ft from the street light. We can set up a proportion using the height of the woman, the height of the street light, and the length of the shadow.

Using the proportion (Shadow Length)/(Woman's Height) = (Street Light Height)/(Distance from Street Light), we can substitute the given values and solve for the Shadow Length.

In this case, Shadow Length = (33 ft/5.5 ft) x 12 ft = 72 ft.

To find the rate of change of the length of the shadow, we can take the derivative of the Shadow Length equation with respect to time. Since the woman is walking towards the street light, the distance from the street light will be decreasing at a rate of 4 ft/s. Consequently, the rate of change of the length of the shadow will be (-4 ft/s) x (33 ft/5.5 ft) = -24 ft/s.