Can someone please help me with this calculus question please

Question

A man starts walking north at 2 ft/s from a point P. Five minutes later a woman starts walking south at 5 ft/s from a point 500 ft due east of P. At what rate are the people moving apart 15 min after the woman starts walking? (Round your answer to two decimal places.)

Answer 1

Since the rates are in ft per second, first convert the times in minutes to seconds.

Let x equal the total distance the man has moved north from his starting point.

Let y equal the total distance the woman has moved south from her starting point.

5 minutes = 5min (60sec/min) = 300 seconds

15 minutes = 900 seconds

Given:

dx/dt = 2ft/sec

dy/dt = 5ft/sec

The woman does not start walking until 5 minutes after the man has started. After 5 minutes (300 seconds) the man is 2(300) or 600 ft north of his starting point. The woman is about to start walking.

To create a right triangle picture the man walking north from the same point as the woman. After 1 second, that leg of the right triangle will be [600+1(2)] + [1(5)] or 607 feet. Another example:

After 10 seconds that leg of the right triangle will be: [600 + 10(2)] + [10(5)] = 670 feet.

The other leg of the right triangle is a constant 500 feet.

By the Pythagorean Theorem:

(x+y)² + 500² = h²

Differentiate with respect to t:

2(x + y) [dx/dt + dy/dt] + 0 = 2h(dh/dt) ; we want to determine dh/dt at the point in time where t = 900 sec (15 minutes after the woman starts walking)

Note that we can divide both sides by 2:

(x + y) [dx/dt + dy/dt] = h(dh/dt)

Now the question becomes what are the values of x, y, h 900 seconds after the woman starts walking?

x = 600 + 900(2) = 2400 ft. The man started 600 feet north and has added on 2 feet for every second.

y = 900(5) = 4500 ft. The woman adds on 5 feet of distance for every second.

(x + y) = 6900 feet which is the vertical leg of the visualized right triangle (of course you drew a diagram).

h = √(500² + 6900²).

6900(2 ft/sec + 5 ft/sec) = √(500² + 6900²) (dh/dt)

dh/dt = 7(6900) / √(500² + 6900) ≈ 6.98 ft/sec

This Desmos graph has a slider on time t. Use the slider from t = 0 to t= 900 to see the dimensions changing.

The last three rows on the graph show how this problem could be solved by setting up a function for the distance between the man and the woman, finding the derivative, and evaluating the derivative at 900.

desmos.com/calculator/qohmda6eiz