Rick's Kayaking Trip: Finding the Speed of the Current

Rick's Kayaking Adventure

Rick paddled up the river, spent the night camping, and then paddled back. He spent a total of 10 hours paddling and the campground was 24 miles away. If Rick kayaked at a speed of 5 mph, what was the speed of the current (in mph)?

Final answer:

To find the speed of the river current affecting Rick's kayak trip, we set up an equation considering his total paddling time and distance to the campground. After setting up the equation based on his speed with and against the current, we determined that the speed of the current is 1 mph.

Explanation:

To solve this problem, we need to understand the concept of relative motion, particularly how the current affects the speed of the kayak. The problem states that Rick paddled up a river and back, spending 10 hours paddling in total and the campground being 24 miles away. We know the speed of the kayak without the current is 5 mph.

Calculating the Speed of the Current

Let's denote the speed of the current as c mph. The speed going upstream against the current will be 5 - c mph, while the speed going downstream with the current will be 5 + c mph. Since the distance to the campground is 24 miles, the time taken to go upstream is 24 / (5 - c) and the time to go downstream is 24 / (5 + c).

We can set up the equation based on the total paddling time:
24 / (5 - c) + 24 / (5 + c) = 10

This is an equation with one unknown (c), which we can solve.

First, find a common denominator and add the two fractions:
(24(5 + c) + 24(5 - c)) / ((5 - c)(5 + c)) = 10

Multiplying out the numerators and combining like terms:
240 + 24c - 24c / (25 - c^2) = 10

The c terms cancel out:

240 / (25 - c^2) = 10

Multiply both sides by (25 - c^2) and divide by 10:
240 = 10 × (25 - c^2)
240 = 250 - 10c^2
10c^2 = 250 - 240
10c^2 = 10

Dividing by 10, we get:
c^2 = 1

Taking the square root:
c = ±1

Since the current cannot have a negative speed, the speed of the current is 1 mph.