Problem 78 - Algebra and Trigonometry Real Mathematics, Real People

Question

Navigation A motorboat traveling with the current takes 40 minutes to travel 20 miles downstream. The return trip takes 60 minutes. Find the speed of the current and the speed of the boat relative to the current, assuming that both remain constant.

Step-by-Step Solution

Answer

The speed of the boat relative to the current is 25 miles per hour and the speed of the current is 5 miles per hour.

Step 1: Defining Variables

Define the variables: Let \( b \) be the speed of the boat in still water (relative to the current) and \( c \) be the speed of the current. Since speed is distance divided by time, downstream, the combined speed of the boat and the current is \( b + c \) miles per hour, and upstream, the effective speed of the boat against the current is \( b - c \) miles per hour.

Step 2: Formulating Equations

Using the provided information, formulate two equations based on the rate-time-distance relationship: For the downstream trip, \( 20 = (b + c) * 40/60 \). For the upstream trip, \( 20 = (b - c) * 60/60 \). Simplifying these equations gives \( b + c = 30 \) and \( b - c = 20 \).

Step 3: Solving for the Variables

Solving these two equations will give the values for \( b \) and \( c \). By adding the two equations, we get \( 2b = 50 \), so \( b = 25 \) miles per hour. Substituting \( b = 25 \) into the first equation, we get \( c = 5 \) miles per hour.

Key Concepts

Rate-Time-Distance Relationship

The rate-time-distance relationship is a fundamental concept in algebra and physics that helps us connect how fast something moves (rate), how far it travels (distance), and how long it takes (time).
This relationship is often expressed with the formula:

• Distance = Rate × Time.

In the context of the problem involving a motorboat, understanding this concept allows us to see that the distance traveled, whether going downstream or upstream, relies heavily on both the speed of the boat and the speed of the current affecting it.
When traveling with the current, the effective speed increases by adding the current's speed to the boat's speed.
On the return trip, against the current, the effective speed decreases, as the current's speed works against the boat.
By expressing these relationships with variables, we can create equations to find unknowns, such as the boat's speed in still water or the current's speed.

Speed of Boat and Current

Understanding the speed of the boat and the current is crucial for solving navigation problems.
In our specific case, we denote the speed of the boat in still water as "b", and the speed of the current as "c".
Downstream, where the boat travels with the current, the effective speed is the sum of the speeds:

• Speed downstream = b + c.

Conversely, when the boat travels upstream against the current, its effective speed is reduced by the current:

• Speed upstream = b - c.

This conceptual differentiation between downstream and upstream speeds allows us to understand and foresee how the boat's travel time alters when it moves with or against a current.
Applying these principles in equations helps in isolating and determining these speeds.

Formulating Equations

To solve a problem like the one involving the boat and the current, we must translate the situation into mathematical equations.

• For the downstream journey, where the boat travels 20 miles in 40 minutes, we use the formula: \[ 20 = (b + c) \times \frac{40}{60}. \]
• For the upstream return trip, covering the same distance in 60 minutes, the formula becomes: \[ 20 = (b - c) \times \frac{60}{60}. \]

These equations arise directly from the rate-time-distance relationship, where we equate the product of rate (speed) and time to distance.
By simplifying these equations, we express the effective speeds simply as: \( b + c = 30 \) and \( b - c = 20 \).
These equations form the mathematical foundation for solving the unknown variables.

Solving Equations

Once we have our equations, it's time to solve them.
The two equations derived were:

• Equation 1: \( b + c = 30 \)
• Equation 2: \( b - c = 20 \)

We can solve these using the method of elimination or substitution.
Adding both equations allows the "c" terms to cancel out, giving us:

• \( 2b = 50 \)

From this, we find that the speed of the boat in still water is \( b = 25 \) miles per hour.
Next, substituting \( b = 25 \) back into the first equation provides:

• \( 25 + c = 30 \).

This gives us the speed of the current as \( c = 5 \) miles per hour.
By systematically solving these equations, we determine the two necessary speeds, providing a clear solution to the navigation problem.