Question

Ben's boat will take 1 1/2 hours longer to go 12 miles up a river than to return. What is the rate of his boat if the rate of the current is 2 miles an hour? 4 mph 5 mph 6 mph

Answer 1

The rate of his boat is:

6 mph

Explanation:

It is given that:

Ben's boat will take 1 1/2 hours longer to go 12 miles up a river than to return.

Let u denote the speed of the boat in still water.

and v denote the speed of the current.

Then the speed of boat upstream= u-v km/h

and speed of boat downstream=u+v km/h

Let t denote the time taken by the boat downstream.

Then the time taken by boat upstream is: t+(3/2) hours

Distance traveled by boat each way is: 12 miles.

Hence, we have:

Speed of boat upstream is:

`(12)/(t+(3)/(2))`

i.e.

`u-v=(12)/(t+(3)/(2))----------(1)`

and

speed of boat downstream i.e.

`u+v=(12)/(t)------------(2)`

On subtracting equation (1) from equation (2) we have:

`2v=(12)/(t)-(12)/(t+(3)/(2))`

Also, we are given :

`v=2\ mph`

i.e.

`2* 2=(12)/(t)-(12)/(t+(3)/(2))`

i.e.

`4=(12* (t+(3)/(2))-12* t)/(t(t+(3)/(2))\\\\i.e.\\\\4=(12t+18-12t)/(t(t+(3)/(2)))\\\\i.e.\\\\4(t(t+(3)/(2)))=18\\\\i.e.\\\\2(t(t+(3)/(2)))=9\\\\i.e.\\\\2t^2+3t=9\\\\i.e.\\\\2t^2+3t-9=0`

i.e.

`2t^2+6t-3t-9=0\\\\i.e.\\\\2t(t+3)-3(t+3)=0\\\\i.e.\\\\(2t-3)(t+3)=0\\\\i.e.\\\\t=(3)/(2)\ or\ t=-3`

But t can't be negative.

Hence, we have:

`t=(3)/(2)`

Hence, from equation (2) we have:

`u+v=(12)/((3)/(2))\\\\i.e.\\\\u+v=(12* 2)/(3)\\\\i.e.\\\\u+v=8\\\\i.e.\\\\u+2=8\\\\i.e.\\\\u=8-2\\\\i.e.\\\\u=6\ mph`

Hence, the answer is: 6 mph

Answer 2

6 mph

Explanation:

Let x mph be the rate of the boat. Going up the river boat has the rate x-2 mph (the current decreases the rate of the boat) and going down the river boat has the rate x+2 mph (the current increases the rate of the boat). Then it will take
`(12)/(x-2)` hours to go up the river and
`(12)/(x+2)` hours to go down the river. Thus,

`(12)/(x-2)-(12)/(x+2)=1(1)/(2),\\ \\(12x+24-12x+24)/((x-2)(x+2))=(3)/(2),\\ \\96=3(x^2-4),\\ \\x^2-4=32,\\ \\x^2=36,\\ \\x=\pm6\ mph.`

The rate of the boat cannot be negative, hence, x=6 mph.