Question

What are the speeds for both parts of a kayak trip, given the first part is 42 miles, the second part is 25 miles at 3 mph slower, and the total trip takes 5 hours?

Answer 1

To find the speeds for both parts of the kayak trip, we can set up the problem using algebra. Let's denote the speed for the first part of the trip as \( x \) mph.

From the problem, we know:
- The first part of the trip is 42 miles at speed \( x \) mph.
- The second part of the trip is 25 miles at speed \( x - 3 \) mph (which is 3 mph slower than the first part).
- The total time for the entire trip is 5 hours.

We can calculate the time taken for each part of the trip using the formula:

\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}}
\]

### Time for the first part:
\[
\text{Time}_1 = \frac{42}{x}
\]

### Time for the second part:
\[
\text{Time}_2 = \frac{25}{x - 3}
\]

### Setting up the total time equation:
The total time for the trip can then be expressed as:

\[
\text{Time}_1 + \text{Time}_2 = 5
\]

Substituting the expressions we have:

\[
\frac{42}{x} + \frac{25}{x - 3} = 5
\]

### Finding a common denominator and solving the equation:

To simplify, we can multiply through by \( x(x - 3) \) to eliminate the denominators:

\[
42(x - 3) + 25x = 5x(x - 3)
\]

Expanding both sides:

\[
42x - 126 + 25x = 5x^2 - 15x
\]

Combining like terms:

\[
67x - 126 = 5x^2 - 15x
\]

Rearranging the equation:

\[
5x^2 - 15x - 67x + 126 = 0
\]

This simplifies to:

\[
5x^2 - 82x + 126 = 0
\]

### Solving the quadratic equation:

We can use the quadratic formula, where \( a = 5 \), \( b = -82 \), and \( c = 126 \):

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

Substituting the values:

\[
x = \frac{82 \pm \sqrt{(-82)^2 - 4(5)(126)}}{2(5)}
\]
\[
= \frac{82 \pm \sqrt{6724 - 2520}}{10}
\]
\[
= \frac{82 \pm \sqrt{4204}}{10}
\]
\[
= \frac{82 \pm 64.8}{10}
\]

Calculating the two potential values for \( x \):

1. \( x = \frac{82 + 64.8}{10} = \frac{146.8}{10} = 14.68 \)
2. \( x = \frac{82 - 64.8}{10} = \frac{17.2}{10} = 1.72 \)

Since \( x \) represents a speed, we take the feasible solution:

**First part speed:**
\[
x \approx 14.68 \text{ mph}
\]

**Second part speed:**
\[
x - 3 \approx 14.68 - 3 = 11.68 \text{ mph}
\]

### Summary:

The speeds for the kayak trip are approximately:
- **First part:** \( 14.68 \) mph
- **Second part:** \( 11.68 \) mph