How can we determine the speed of the stream if a motorboat travels at 24 km/h in still water and takes an hour longer to go 32 km upstream compared to its return trip downstream?

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To solve this problem, we start by setting up the necessary parameters from the information given.

Let:

  • Speed of the motorboat in still water = 24 km/h
  • Speed of the stream = x km/h

When the motorboat is going upstream, the effective speed is decreased due to the current of the stream, while downstream, the speed increases. Therefore, we have:

  • Upstream speed = 24 – x
  • Downstream speed = 24 + x

Next, we can calculate the time taken to travel each distance:

  • Time taken to go upstream: Timeup = \frac{Distance}{Speed} = \frac{32}{24 - x}
  • Time taken to go downstream: Timedown = \frac{Distance}{Speed} = \frac{32}{24 + x}

According to the problem, the time taken to go upstream is 1 hour more than the time taken to return downstream. Thus, we set up the equation:

Timeup = Timedown + 1

Substituting the expressions for the times we calculated earlier:

\frac{32}{24 - x} = \frac{32}{24 + x} + 1

Next, we can eliminate the fractions by multiplying through by the product of the denominators:

(24 - x)(24 + x)

Thus, we obtain:

32(24 + x) = 32(24 - x) + (24 - x)(24 + x)

Expanding both sides gives:

768 + 32x = 768 - 32x + (576 - x^2)

Reorganizing the equation leads to:

32x + 32x + x^2 = 576

Combine like terms:

x^2 + 64x - 576 = 0

This is a quadratic equation in standard form. We can apply the quadratic formula to find the values of x:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Where a = 1, b = 64, and c = -576:

b^2 - 4ac = 64^2 - 4(1)(-576) = 4096 + 2304 = 6400

Now, we substitute back into the quadratic formula:

x = \frac{-64 \pm 80}{2}

Calculating both potential solutions gives us:

  • x = \frac{16}{2} = 8
  • x = \frac{-144}{2} = -72 (not possible, since speed cannot be negative)

So, the speed of the stream is 8 km/h.

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