Acamper paddles a canoe 2 miles downstream in a river that has a 2-mile-per-hour current. to return to camp, the canoeist travels upstream on a different branch of the river. it is 4 miles long and has a 1-mile-per-hour current. the total trip (both ways) takes 3 hours. find the average speed of the canoe in still water. time = distance/rate

The average speed of the canoe in still water is 2.56 miles/hour.

Step-by-step explanation:

Step 1 :

According to the given information:

Canoe speed downstream = 2 miles/hour

Canoe speed upstream = 1 mile/hour  

Distance with downstream = 2 miles

Distance with upstream = 4 miles

Let the average speed be ‘x’

Then, Speed with current = x+2 miles/hour

And, speed against current = x-1 miles/hour

Total time = 3 hours

Therefore,

time with current (downstream) = 2 / x + 2

time against current (upstream) = 4 / x - 1

Total time = (2 / x + 2) + (4 / x - 1)

Step 2:

Now, put the value of total time and solve:

Total time = (2 / x + 2) + (4 / x - 1)

3 = (2 / x + 2) + (4 / x - 1)

3 = (2(x - 1) + 4(x + 2)) / (x + 2)(x - 1)

3 = (2x - 2 + 4x + 8) / (x² + x - 2)

3(x² + x - 2) = 6x + 6

x² - x - 4 = 0

Step 3:

Solving the quadratic equation:

x = (-(-1) ±√[ (-1)² - 4(1)(-4) ] ) / 2(1)

x = 1 ± √(17) / 2

⇒ x₁ = 2.56,  x₂ = -1.56

Step 4:

Since an average speed cannot be negative, the answer will be x₁ = 2.56.

Therefore, the average speed of the canoe in still water is 2.56 miles/hour.

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step-by-step explanation: