Problem 44 - Intermediate Algebra

Question

Kayaking. A kayaker can travel 1.2 miles downstream in the same time it takes him to go 0.4 miles upstream. If the river current flows at \(2 \mathrm{mph}\), what is the kayaker's speed in still water?

Step-by-Step Solution

Answer

The kayaker's speed in still water is 4 mph.

Step 1: Define Variables

Let's define the variable: Let \( x \) be the kayaker's speed in still water (in mph). The speed of the river current is given as \( 2 \) mph.

Step 2: Determine Downstream and Upstream Speeds

When traveling downstream, the kayaker's speed is the sum of his speed in still water and the river's current speed, which is \( x + 2 \) mph. When traveling upstream, his speed becomes \( x - 2 \) mph because the current is against him.

Step 3: Write the Time Equations

Based on the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \), the time to travel downstream is \( \frac{1.2}{x+2} \) hours and the time to travel upstream is \( \frac{0.4}{x-2} \) hours. We know these two times are equal.

Step 4: Set Up the Equation

Since the travel times downstream and upstream are equal, we can equate these two expressions: \[ \frac{1.2}{x+2} = \frac{0.4}{x-2} \]

Step 5: Solve the Equation

Cross-multiply to solve for \( x \): \( 1.2(x - 2) = 0.4(x + 2) \). Simplify and solve: \( 1.2x - 2.4 = 0.4x + 0.8 \). Bringing all terms involving \( x \) to one side yields: \( 1.2x - 0.4x = 0.8 + 2.4 \), which simplifies to \( 0.8x = 3.2 \). Dividing both sides by 0.8 gives \( x = 4 \).

Step 6: Verify the Solution

Substitute \( x = 4 \) back into the time expressions to verify the solution. The time downstream is \( \frac{1.2}{4+2} = \frac{1.2}{6} = 0.2 \) hours. The time upstream is \( \frac{0.4}{4-2} = \frac{0.4}{2} = 0.2 \) hours. Since both times are equal, the solution is verified.

Key Concepts

Downstream and Upstream Speeds

Understanding how speeds change when kayaking is crucial. The river's current can either help you speed up or slow you down depending on your direction. When you're going downstream, the current adds to your speed. Imagine you're kayaking and your speed in calm water is 4 mph. If the river’s current is 2 mph, then downstream, you'll be moving at a speed of \(4 + 2 = 6 \) mph. You're not doing more work; the current is doing some for you.

• Downstream Speed: Determined by adding the kayaker's speed to the river current.
• Upstream Speed: The speed of the current is subtracted from the kayaker's speed.

However, when traveling upstream, you have to work against the current. Now, only part of your overall speed is effective in getting you to your destination. So, if you kayak at 4 mph, you minus the current's push against you; your speed becomes \(4 - 2 = 2 \) mph. It takes more time and effort. Thus, understanding how downstream and upstream speeds work is vital for calculating total travel time in water.

Time Equations in Algebra

Time equations are handy tools when it comes to analyzing movement problems like this one. You can think of time as the link that connects distance and speed. The basic formula here is \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \). If you know any two of these variables, you can easily solve for the third. Imagine a kayaker travels 1.2 miles downstream and 0.4 miles upstream. By using the speeds calculated for both scenarios, we can set up time equations.

• For the downstream trip, the time is \( \frac{1.2}{x+2} \), where \( x \) is the kayaker's speed in calm water.
• For the upstream trip, the time is \( \frac{0.4}{x-2} \).

In this exercise, the two times are the same, meaning our equations balance. That's key for setting up the equation you need to solve. This balance allows us to equate them and solve for the unknown variable, the speed in still water.

Solving Linear Equations

Solving linear equations is the process where we find the value of the variable that makes the equation true. In our case, both the downstream and upstream scenarios give us equations based on time. Once we know they are equal, we can set them against each other. Our equation becomes \( \frac{1.2}{x+2} = \frac{0.4}{x-2} \). This kind of equation needs a little rearranging. Cross-multiplying helps to get the variables out of the denominator and onto a single line.

• First, multiply both sides: \( 1.2(x - 2) = 0.4(x + 2) \).
• Then, distribute and simplify each side: \( 1.2x - 2.4 = 0.4x + 0.8 \).

Collect all terms with \( x \) on one side and constants on the other. Simplifying further gives us \( 0.8x = 3.2 \). Divide both sides by 0.8 to find \( x \), which is 4. This tells us the kayaker's speed in still water. To verify, substitute back into the original time formulas to ensure both times are equal, confirming our solution is correct. This method of solving provides accuracy and confirms the correctness of the variable's value.