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Chapter 3: Problem 39

In his motorboat, Bill travels upstream at top speed to his favorite fishing spot, a distance of \(36 \mathrm{mi}\), in 2 hr. Returning, he finds that the trip downstream, still at top speed, takes only \(1.5 \mathrm{hr}\). Find the rate of Bill's boat and the rate of the current. Let \(x=\) the rate of the boat and \(y=\) the rate of the current.

Short Answer

Expert verified
The rate of the boat is 21 mph, and the rate of the current is 3 mph.

Step by step solution

01

- Define variables

Let the speed of the boat in still water be denoted by \( x \) (in mph), and let the speed of the current be denoted by \( y \) (in mph).
02

- Write the equation for upstream travel

When traveling upstream, the current works against the boat. The effective speed of the boat upstream is \( x - y \). Given that the distance is 36 miles and it takes 2 hours, we can write the equation: \[ 36 = 2(x - y) \].
03

- Simplify the upstream equation

Divide both sides of the equation by 2 to solve for \( x - y \): \[ x - y = 18 \].
04

- Write the equation for downstream travel

When traveling downstream, the current works with the boat. The effective speed of the boat downstream is \( x + y \). Given that the distance is 36 miles and it takes 1.5 hours, we can write the equation: \[ 36 = 1.5(x + y) \].
05

- Simplify the downstream equation

Divide both sides of the equation by 1.5 to solve for \( x + y \): \[ x + y = 24 \].
06

- Solve the system of equations

We now have two equations: \[ x - y = 18 \] and \[ x + y = 24 \]. Add these equations to eliminate \( y \): \[ (x - y) + (x + y) = 18 + 24 \], which simplifies to \[ 2x = 42 \], so \[ x = 21 \].
07

- Solve for y

Substitute \( x = 21 \) back into one of the original equations, \( x + y = 24 \): \[ 21 + y = 24 \]. Solving for \( y \): \[ y = 3 \].
08

- State the final answer

The rate of the boat in still water is \( 21 \ \text{mph}\) and the rate of the current is \( 3 \ \text{mph}\).

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Key Concepts

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

systems of equations
A system of equations is a set of two or more equations with the same variables. In this exercise, we used a system of two linear equations to find the unknown speeds.
upstream and downstream problems
Upstream and downstream problems involve calculating the effective speed of a boat traveling in a river with a current. These problems are a classic example of how systems of linear equations are used to solve real-world problems.:
linear equations
Linear equations are algebraic expressions that describe a straight-line relationship between two variables. They are typically in the form ax + by = c.

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Most popular questions from this chapter

Solve each system. $$\begin{array}{l}4 x-z=-6 \\\\\frac{3}{5} y+\frac{1}{2} z=0 \\\\\frac{1}{3} x+\frac{2}{3} z=-5\end{array}$$

Determine whether the given ordered pair is a solution of the given system. $$ \begin{array}{l} 2 x-y=8 \\ 3 x+2 y=20 \end{array} ; \quad(5,2) $$

Solve each system using the substitution method. If a system is inconsistent or has dependent equations, say so. $$ \begin{array}{l} x=5 y \\ 5 x-25 y=5 \end{array} $$

A motor scooter travels 20 mi in the same time that a bicycle travels \(8 \mathrm{mi}\). If the rate of the scooter is 5 mph more than twice the rate of the bicycle, find both rates.

Complete each statement. If solving a system leads to a false statement such as \(0=3,\) the solution set is _____________.
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