A sailor can row a boat 8 km downstream and return back to the starting point in 1 hr. 40 min. If the speed of the stream is 2 km/hr, find the speed of the boat in still water?

To find the speed of the boat in still water, we first need to break down the problem. The total time taken for the round trip is 1 hour and 40 minutes, which is equivalent to 100 minutes or 100/60 = 5/3 hours.

Understanding the Journey

The sailor rows 8 km downstream and then returns upstream. The speed of the stream is given as 2 km/hr.

Calculating Effective Speeds

Let the speed of the boat in still water be b km/hr. The effective speeds are:

• Downstream speed = b + 2 km/hr
• Upstream speed = b - 2 km/hr

Time for Each Leg of the Journey

The time taken to row downstream and upstream can be expressed as:

• Time downstream = Distance / Speed = 8 / (b + 2)
• Time upstream = Distance / Speed = 8 / (b - 2)

Setting Up the Equation

The total time for the round trip is the sum of the downstream and upstream times:

8 / (b + 2) + 8 / (b - 2) = 5/3

Solving the Equation

To solve for b, we first find a common denominator:

(b + 2)(b - 2)

Multiplying through by the common denominator gives:

8(b - 2) + 8(b + 2) = (5/3)(b + 2)(b - 2)

Simplifying this leads to:

8b - 16 + 8b + 16 = (5/3)(b^2 - 4)

16b = (5/3)(b^2 - 4)

Rearranging the Equation

Multiplying everything by 3 to eliminate the fraction:

48b = 5(b^2 - 4)

48b = 5b^2 - 20

5b^2 - 48b - 20 = 0

Finding the Roots

Using the quadratic formula b = [-B ± √(B² - 4AC)] / 2A, where A = 5, B = -48, and C = -20:

b = [48 ± √((-48)² - 4 × 5 × (-20))] / (2 × 5)

b = [48 ± √(2304 + 400)] / 10

b = [48 ± √2704] / 10

b = [48 ± 52] / 10

Calculating the Speed

This gives two possible solutions:

• b = 10 km/hr (valid speed)
• b = -0.4 km/hr (not valid)

Final Answer

The speed of the boat in still water is 10 km/hr.