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Chapter 7: Problem 20

Set up the equation you would use to solve each problem. Do not actually solve the equation. Janet Sturdy can row 4 mph in still water. She takes as long to row 8 mi upstream as 24 mi downstream. How fast is the current? (Let \(x=\) rate of the current.)

Short Answer

Expert verified
Set up the equation: \( \frac{8}{4 - x} = \frac{24}{4 + x} \).

Step by step solution

01

- Define the variables

Let Janet's rowing speed in still water be 4 mph. Let the speed of the current be denoted by \( x \) mph.
02

- Determine effective speeds

When rowing upstream, Janet's effective speed will be \(4 - x\) mph because she is going against the current. When rowing downstream, her effective speed will be \(4 + x\) mph because she is moving with the current.
03

- Write the time formulas

Use the formula \( \text{time} = \frac{\text{distance}}{\text{speed}} \). The time taken to row 8 miles upstream is \( \frac{8}{4 - x} \) hours. The time taken to row 24 miles downstream is \( \frac{24}{4 + x} \) hours.
04

- Set up the equation

Since the time taken to row upstream and downstream is the same, set the equations equal to each other: \( \frac{8}{4 - x} = \frac{24}{4 + x} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rate Problems
Rate problems involve finding the speed, distance, or time when one or more of these quantities are known. To solve these problems, you often use the fundamental formula \[ \text{time} = \frac{\text{distance}}{\text{speed}} \]. This formula helps express the relationship between time, distance, and speed. For instance, if you know the distance traveled and the speed, you can calculate the time it takes to cover that distance.

In rate problems, the given values can sometimes vary due to different conditions, like moving upstream or downstream in the case of rowing in a river. Keep in mind that these variations need to be accounted for when setting up the equations.
Upstream and Downstream Motion
When dealing with the motion of objects in water, the current adds complexity to the problem. Understand that:
  • Upstream means moving against the current.
  • Downstream means moving with the current.

Given Janet's rowing speed of 4 mph in still water, her effective speed changes due to the current.

  • Upstream speed: 4 - x mph (where x is the speed of the current)
  • Downstream speed: 4 + x mph

As mentioned in the exercise, Janet takes the same time to row 8 miles upstream as 24 miles downstream. This means the different effective speeds must be taken into account to match the times. Using the formula \[ \text{time} = \frac{\text{distance}}{\text{speed}} \], you can set up the equations to represent these scenarios.
Equation Setup
Setting up equations correctly is crucial in solving rate problems, especially ones involving motion in water. Let鈥檚 go through the steps:

  • First, identify and define all the variables involved. Here, Janet鈥檚 rowing speed in still water is 4 mph, and we denote the current鈥檚 speed as x mph.
  • Then, determine the effective speeds for both upstream and downstream motions.
  • For the upstream motion, the effective speed is 4 - x mph, and for the downstream motion, it is 4 + x mph.

The time taken to travel upstream for 8 miles is \[ \frac{8}{4 - x} \] hours, and the time taken to go downstream for 24 miles is \[ \frac{24}{4 + x} \] hours. Since these times are equal, we can set the equations equal to each other:

\[ \frac{8}{4 - x} = \frac{24}{4 + x} \]

This setup allows us to solve for x , the speed of the current.

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