Calculating Boat Water Speed

To find the speed of the water, we first need to calculate the downstream and upstream speeds of the boat.

Downstream Speed Calculation

The boat travels 288 km downstream in 32 hours. The downstream speed ($v_d$) is calculated using the formula:

$Speed = Distance / Time$

So, the downstream speed is:

$v_d = \frac{288 \text{ km}}{32 \text{ hours}} = 9 \text{ kmph}$

Upstream Speed Calculation

The boat travels 217 km upstream in 31 hours. The upstream speed ($v_u$) is calculated similarly:

$v_u = \frac{217 \text{ km}}{31 \text{ hours}} = 7 \text{ kmph}$

Water Speed Determination

Let the speed of the boat in still water be $v_b$ and the speed of the water (stream) be $v_w$. The formulas relating these speeds are:

• Downstream Speed: $v_d = v_b + v_w$
• Upstream Speed: $v_u = v_b - v_w$

To find the speed of the water ($v_w$), we can subtract the upstream speed equation from the downstream speed equation:

$v_d - v_u = (v_b + v_w) - (v_b - v_w)$

$v_d - v_u = v_b + v_w - v_b + v_w$

$v_d - v_u = 2v_w$

Rearranging the formula to solve for $v_w$:

$v_w = \frac{v_d - v_u}{2}$

Now, substitute the calculated downstream and upstream speeds:

$v_w = \frac{9 \text{ kmph} - 7 \text{ kmph}}{2}$

$v_w = \frac{2 \text{ kmph}}{2}$

$v_w = 1 \text{ kmph}$

Therefore, the speed of the water is 1 kmph.