Problem 66 - Precalculus: Mathematics for Calculus

Question

Boat Speed \(A\) boat on a river travels downstream between two points, \(20 \mathrm{mi}\) apart, in \(1 \mathrm{h}\). The return trip against the current takes \(2 \frac{1}{2} \mathrm{h} .\) What is the boat's speed, and how fast does the current in the river flow?

Step-by-Step Solution

Answer

The boat's speed is 14 mph and the current flows at 6 mph.

Step 1: Define Variables

Let the speed of the boat in still water be \( b \) mph and the speed of the current be \( c \) mph. When the boat is moving downstream, its effective speed is \( b + c \) mph. When moving upstream, its effective speed is \( b - c \) mph.

Step 2: Formulate Equations for Downstream and Upstream

For the downstream journey, we have:\[ \text{Distance} = \text{Speed} \times \text{Time} \]Given, \( 20 = (b + c) \times 1 \).For the upstream journey, we have:\[ 20 = (b - c) \times 2.5 \]

Step 3: Solve the Downstream Equation

From the downstream equation \( 20 = b + c \), we can solve for \( b + c = 20 \). This means \( b + c = 20 \) mph.

Step 4: Solve the Upstream Equation

From the upstream equation \( 20 = (b - c) \times 2.5 \), solving for \( b - c \):\[ 20 = 2.5b - 2.5c \]\[ 8 = b - c \] (after dividing the entire equation by 2.5).

Step 5: Solve the System of Equations

Now solve the system of equations:\[ b + c = 20 \]\[ b - c = 8 \]Add the equations:\[(b + c) + (b - c) = 20 + 8 \rightarrow 2b = 28 \rightarrow b = 14 \]Substitute \( b = 14 \) back into the first equation:\[ 14 + c = 20 \rightarrow c = 6 \]

Step 6: Interpret the Results

The boat's speed in still water is \( b = 14 \) mph, and the current speed is \( c = 6 \) mph.

Key Concepts

Downstream and Upstream Travel

When a boat moves along a river, the speed of the stream affects the overall speed of the journey. This is because the current of the river can either aid or hinder the movement of the boat. When traveling **downstream**, the current of the river helps the boat, making the effective speed faster. For a boat with a speed of \( b \) mph in still water, the downstream speed becomes \( b + c \) mph, where \( c \) represents the speed of the current.On the other hand, when the boat travels **upstream**, it is moving against the current, which slows it down. The effective upstream speed is then \( b - c \) mph. Understanding these differences is crucial for setting up the correct mathematical equations to solve speed-related problems. This concept helps us account for the additional energy and time required when going upstream, as well as the time saved when going downstream.

System of Equations

To solve the boat speed problem, setting up a system of equations is essential. In our problem, we are given two conditions: one for downstream travel and one for upstream travel.By translating these scenarios into mathematical equations, we can better analyze the relationships between the various speeds involved:- **Downstream Equation**: With the effective speed being \( b + c \), and the trip taking 1 hour, the equation is \( 20 = (b + c) \times 1 \). This simplifies to \( b + c = 20 \).- **Upstream Equation**: For the return trip, the effective speed is \( b - c \) and it takes 2.5 hours, leading to \( 20 = (b - c) \times 2.5 \). Upon simplification, \( b - c = 8 \).Together, these two equations form a system:

• \( b + c = 20 \)
• \( b - c = 8 \)

By solving these equations simultaneously, we can find the values of \( b \) and \( c \), representing the boat's speed in still water and the current's speed, respectively.

Distance-Speed-Time Relationship

The relationship between distance, speed, and time is fundamental in understanding travel problems like the boat speed problem. This basic concept states:\[ \text{Distance} = \text{Speed} \times \text{Time} \]This formula helps you make sense of how changes in one variable affect the others. For instance, if the speed of travel increases, the time taken to cover the same distance decreases, assuming the distance remains constant.In our boat speed problem, both downstream and upstream travels adhere to this relationship:- For the downstream journey: the distance covered is 20 miles at an effective speed of \( b + c \) mph in 1 hour.- For the upstream voyage: the same distance of 20 miles at the speed of \( b - c \) mph takes 2.5 hours.Using these scenarios, you apply the distance formula twice to generate the necessary equations to find the unknown speeds of the boat and the river current.