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You and a friend are riding your bikes to a restaurant that you think is east and that your friend thinks is north. You both leave from the same starting point, with you riding 16 miles per hour east and your friend riding 12 miles per hour north. After you travel 4 miles, at what rate is the distance between you changing?

Respuesta :

Answer:

20 miles per hour

Explanation:

Let the east direction is denoted by x and the north direction is denoted by y.

so, dx / dt = 16 miles per hour

dy / dt = 12 miles per hour

x = 4 miles

Let the distance between both the friends be s.

Time taken by you to travel 4 mile, t = distance / speed = 4 / 12 = 1/4 hour

Distance traveled by your friend in time t, y = speed x time = 12 x 1/4 = 3 miles

Use Pythagorean theorem,

x^2 + y^2 + s^2     ..... (1)

Substitute, x = 4 mile, y = 3 mile

s^2 = 16 + 9 = 25

s = 5 miles

Differentiate equation (1) with respect to t, we get

2x dx/dt + 2y dy/dt = 2s ds/dt

4 (16) + 3 (12) = 5 ds/dt

ds/dt = 20 miles per hour

Thus, the distance between you and your friend is changing with a rate of 20 miles per hour.

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