We can rewrite the function f(x) to be a function of time, where x =5t.
f(x) = (3/400)(x)(40 - x)
f(5t) = (3/400)(5t)(40 - 5t) . . . . substituting 5t for x
Simplifying, we have the current speed as a function of the time spent in the river.
f(t) = (3/16)·t·(8 -t)
The area under this curve is the product of speed and time, thus is distance. To find the distance, we integrate over the time it takes to get from one bank to the other, that is from t=0 to t=8.
∫[0, 8] (3/16)·(8t -t²)dt = 3/16·(8·8²/2 -8³/3) = 8³/32 = 16 . . . . meters
The boat ends up 16 meters downriver.
f(x) = (3/400)(x)(40 - x)
f(5t) = (3/400)(5t)(40 - 5t) . . . . substituting 5t for x
Simplifying, we have the current speed as a function of the time spent in the river.
f(t) = (3/16)·t·(8 -t)
The area under this curve is the product of speed and time, thus is distance. To find the distance, we integrate over the time it takes to get from one bank to the other, that is from t=0 to t=8.
∫[0, 8] (3/16)·(8t -t²)dt = 3/16·(8·8²/2 -8³/3) = 8³/32 = 16 . . . . meters
The boat ends up 16 meters downriver.
