Determining the Speed of a Boat in Still Water and the Rate of the Current

In this article, we will explore a problem that involves determining the speed of a boat in still water and the rate of the current. By understanding this problem, you can apply the principles of distance, speed, and time to solve similar questions. Let's dive into the details and solve the problem using mathematical equations.

The Problem

A boat travels 160 miles upstream in 5 hours and returns the same 160 miles downstream in 4 hours. We need to find the speed of the boat in still water and the rate of the current.

Variables and Given Information

We will use the following variables:

b: Speed of the boat in still water in miles per hour. c: Speed of the current in miles per hour.

Given:

Upstream: Distance 160 miles Time 5 hours Effective speed upstream b - c Downstream: Distance 160 miles Time 4 hours Effective speed downstream b c

Solving the Equations

Using the formula Distance Speed × Time, we can write the following equations:

Upstream:
160 (b - c) × 5
Simplifying, we get:
b - c 32
Downstream:
160 (b c) × 4
Simplifying, we get:
b c 40

Solving the system of equations:

Adding the two equations: (b - c) (b c) 32 40
Simplifying, we get:
2b 72
Solving for b: b frac{72}{2} 36 (Speed of the boat in still water)

Substituting b 36 into the first equation:

36 - c 32 Solving for c: c 36 - 32 4 (Speed of the current)

Final Results:

Speed of the boat in still water: 36 miles per hour Speed of the current: 4 miles per hour

Additional Examples for Practice

Let's apply the same method to solve a couple of similar problems.

Example 1

162/3 - 132/3 / 2 54 - 44/2 5 miles/hour - the rate of the current

54 – 5 44 5 49 miles/hour - the rate of the boat in still water

Example 2

x - y 152/4 38

x y 224/4 56

Add 2x 94

x speed in still water 47 km/h

y rate of current 9 km/h

Example 3

Let the speed of the boat in still water be X and the speed of the current be Y

Upstream: X - Y 52Km/h X 52Km/h - Y Downstream: X Y 80Km/h X Y 80Km/h - Y Y 80Km/h 2Y 80Km/h - 52Km/h 28Km/h Y 14Km/h X 52Km/h - 14Km/h 66Km/h The speed of the boat in still water is 66Km/h The speed of the current is 14Km/h

Conclusion

By understanding the basic principles of distance, speed, and time, and by applying algebraic methods, we can solve problems involving the speed of a boat in still water and the rate of the current. This article provides a clear step-by-step guide to solve such problems, enhancing your problem-solving skills in mathematics.