Question

A street light is at the top of a 17 ft tall pole. A woman 5.5 ft tall walks away from the pole with a speed of 6 ft/sec along a straight path. How fast is the shadow lengthening when she is 40 ft from the base of the pole? Round answers to 3 decimal places.

Answer 1

The problem involves using similar triangles and related rates calculus to find the rate at which the shadow cast by a woman walking away from a street light lengthens. The woman's distance from the pole and the height of both the woman and the pole provide the needed proportions. The rate of change of the shadow's length (dy/dt) can be determined by differentiating those proportions with respect to time.

Step-by-step explanation:

You are being asked to determine how fast the shadow of the woman lengthens as she walks away from the light pole. Since the woman is moving at a constant speed, this problem can be approached using the principles of similar triangles and the concept of related rates in calculus.

Let's denote the woman's distance from the pole as x (which is 40 ft at the moment we are interested in) and her shadow's length as y. According to the information provided, the woman is 5.5 ft tall and the pole is 17 ft tall. By setting up the proportion of the woman's height to the pole's height, which is the same as her distance from the pole to the entire distance from the pole to the end of the shadow (x + y), we have:

5.5 / 17 = x / (x + y)

We are interested in finding dy/dt, the rate at which the length of the shadow is changing with respect to time. Differentiating both sides of the equation with respect to time (t), we can find this rate.