Question

A motorboat whose speed is 18 kmph in still water takes 1 hr 30 min more to go 36 km upstream than to return downstream to the same spot. Find the speed of the stream.

Answer 1

Given: Speed of motorboat in still water = 18 km/h.

Distance = 36 km.

Time difference (upstream – downstream) = 1 hr 30 min = 1.5 h.

Step-wise calculation:

1. Let speed of stream = x km/h.

2. Upstream speed = 18 – x.

Downstream speed = 18 + x.

3. Time upstream = `36/(18 - x)`.

Time downstream = `36/(18 + x)`.

4. Given: `36/(18 - x) - 36/(18 + x) = 1.5`.

5. Factor: `36((18 + x) - (18 - x))/((18 - x)(18 + x)) = 1.5`

⇒ `(36 xx (2x))/(324 - x^2) = 1.5` 

⇒ 72x = 1.5(324 – x²) = 486 – 1.5x² 

⇒ 1.5x² + 72x – 486 = 0 

⇒ Multiply by 2: 3x² + 144x – 972 = 0 

⇒ Divide by 3: x² + 48x – 324 = 0

6. Solve quadratic:

Discriminant = 48² – 4(1)(–324) 

= 2304 + 1296

= 3600, `sqrt(D) = 60`

`x = (-48 ± 60)/2` 

⇒ `x = (12)/2 = 6` or `x = (-108)/2 = -54`.

7. Reject negative root.

So, x = 6 km/h.

Speed of the stream = 6 km/h.