Question

Running at maximum speed, it takes a boat times as long to go 10 miles upstream as it does to go 10 miles downstream. If the current is 4 mph, find the speed of the boat in still water.

Answer 1

Note: The question states the time to go upstream is a number of times (not explicitly written) the time to go downstream. We'll assume a general number N

Answer:

`\displaystyle v_b=(N+1)/(N-1)(4\ mph)`

Step-by-step explanation:

Relative Speed

If a boat is going upstream against the water current, the true speed of motion is
`v_b-v_w`, being
`v_b` the speed of the boat and
`v_w` the speed of the water. If the boat is going downstream, the true speed becomes
`v_b+v_w`.

The question states the time to go upstream is a number of times N (not explicitly written) the time to go downstream. The speed of an object is computed as

`\displaystyle v=(x)/(t)`

Where x is the distance traveled and t the time taken for that. The time can be computed by

`\displaystyle t=(x)/(v)`

If
`t_u` is the time for the upstream travel and
`t_d` is the time for the downstream travel, then

`t_u=Nt_d`

Siince the same distance x= 10 miles is traveled in both cases:

`\displaystyle (10)/(v_b-v_w)=N(10)/(v_b+v_w)`

Simplifying and rearrangling

`v_b+v_w=N(v_b-v_w)`

Operating

`v_b+v_w=Nv_b-Nv_w`

Solving for
`v_b`

`\displaystyle v_b=(N+1)/(N-1)v_w`

`If\ N=2,\ v_w=4\ mph`

`\displaystyle v_b=(3)/(1)(4)=12\ mph`

If N=3

`\displaystyle v_b=(4)/(2)v_w=2(4)=8\ mph`

We can use the required value of N to compute the speed of the boat as explained