A runner completes a 16 mile trip, running 8 miles uphill and 8 miles downhill in a total of 4 hours, if his uphill speed is 3 mph slower than his downhill speed, find his downhill speed.
add up the times
time = distance/speed
If his downhill speed is s,
8/s + 8/(s-3) = 4
s = 6
check:
8 miles @ 3mph = 8/3 hr
8 miles @ 6mph = 4/3 hr
total: 4 hr
To find the runner's downhill speed, let's break down the information provided.
Let's assume the runner's downhill speed is x mph. We are given that the uphill speed is 3 mph slower than the downhill speed, so the uphill speed would be (x - 3) mph.
Next, we know that the runner ran 8 miles uphill and 8 miles downhill, so the time it took to run uphill can be calculated by dividing the distance by the speed: 8 miles / (x - 3) mph.
Similarly, the time it took to run downhill can be calculated by dividing the distance by the speed: 8 miles / x mph.
According to the problem, the total time taken for the entire 16 mile trip is 4 hours. So we can add up the times for uphill and downhill running and set it equal to 4: (8 miles / (x - 3) mph) + (8 miles / x mph) = 4.
To solve this equation, we can multiply through by x(x - 3) to eliminate the denominators:
8x + 8(x - 3) = 4x(x - 3).
Simplifying further:
8x + 8x - 24 = 4x^2 - 12x.
Rearranging the terms:
4x^2 - 28x + 24 = 0.
Dividing through by 4:
x^2 - 7x + 6 = 0.
This quadratic equation can be factored:
(x - 6)(x - 1) = 0.
Therefore, x can be either 6 or 1. However, in the problem, we are looking for the downhill speed, so the answer is 6 mph.