One car travel South at 64 mi/h and the other travels west at 48 mi/h. The distance between the cars increases with rate at 80 mi/h.
To find the rate of change, we need to find the derivative of the variables with respect to time.
Let:
p = distance between 2 cars
q = distance between car 1 and the start point
r = distance between car 2 and the start point
Using the Pythagorean Theorem:
p² = q² + r²
Take the derivative with respect to time:
2p dp/dt = 2q dq/dt + 2r dr/dt
dq/dt = speed of car 1 = 64 mi/h
dr/dt = speed of car 2 = 48 mi/h
The distance of car 1 and car 2 from the start point after 4 hours:
q = 64 x 4 = 256 miles
r = 48 x 4 = 192 miles
Using the Pythagorean theorem:
p² =256² + 192²
p = 320 miles
Hence,
2p dp/dt = 2q dq/dt + 2r dr/dt
p dp/dt = q dq/dt + r dr/dt
320 x dp/dt = 256 x 64 + 192 x 48
dp/dt = 80
Hence, the distance between the cars increases with rate at 80 mi/h
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