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Math  /  Algebra

Question18. Two friends can paddle a canoe at a rate of 6 km/h6 \mathrm{~km} / \mathrm{h} in still water. It takes them 1 h to paddle 2 km up a river and back again. Find the speed of the current.

Studdy Solution

STEP 1

1. The speed of the canoe in still water is 6 km/h6 \, \text{km/h}.
2. The total time taken to paddle 2 km upstream and 2 km downstream is 1 hour.
3. We need to find the speed of the current.

STEP 2

1. Define variables for the speed of the current and the effective speeds upstream and downstream.
2. Write equations for the time taken to travel upstream and downstream.
3. Set up an equation for the total time and solve for the speed of the current.

STEP 3

Define variables.
Let c c be the speed of the current in km/h.
The effective speed upstream is 6−c 6 - c km/h.
The effective speed downstream is 6+c 6 + c km/h.

STEP 4

Write equations for the time taken to travel upstream and downstream.
The time taken to travel upstream is given by:
Time upstream=Distance upstreamEffective speed upstream=26−c \text{Time upstream} = \frac{\text{Distance upstream}}{\text{Effective speed upstream}} = \frac{2}{6 - c}
The time taken to travel downstream is given by:
Time downstream=Distance downstreamEffective speed downstream=26+c \text{Time downstream} = \frac{\text{Distance downstream}}{\text{Effective speed downstream}} = \frac{2}{6 + c}

STEP 5

Set up an equation for the total time and solve for the speed of the current.
The total time for the round trip is 1 hour, so:
26−c+26+c=1 \frac{2}{6 - c} + \frac{2}{6 + c} = 1

STEP 6

Solve the equation.
First, find a common denominator and combine the fractions:
2(6+c)+2(6−c)(6−c)(6+c)=1 \frac{2(6 + c) + 2(6 - c)}{(6 - c)(6 + c)} = 1
Simplify the numerator:
12+2c+12−2c36−c2=1 \frac{12 + 2c + 12 - 2c}{36 - c^2} = 1
2436−c2=1 \frac{24}{36 - c^2} = 1
Cross-multiply to solve for c2 c^2 :
24=36−c2 24 = 36 - c^2
Rearrange the equation:
c2=36−24 c^2 = 36 - 24
c2=12 c^2 = 12
Take the square root of both sides:
c=12 c = \sqrt{12}
c=23 c = 2\sqrt{3}
The speed of the current is 23 km/h 2\sqrt{3} \, \text{km/h} .

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