Problem 26 - Algebra A Combined Function

Question

The Jonathan Schultz family took a canoe 10 miles down the Allegheny River in \(1 \frac{1}{4}\) hours. After lunch it took them 4 hours to return. Find the rate of the current. Let \(x=\) rate the family can row in still water and \(y=\) rate of the current

Step-by-Step Solution

Answer

The rate of the current is 2.75 miles per hour.

Step 1: Understand the problem

We have two scenarios: Jonathan Schultz and his family are traveling downstream and upstream. We need to find the rate of the current.

Step 2: Define the equations based on given information

For downstream, the effective rate is the sum of the rowing rate in still water and the current rate: \(x + y\). They traveled 10 miles in \(1 \frac{1}{4}\) hours, which is \(\frac{5}{4}\) hours. So, the equation is \(10 = (x+y)\frac{5}{4}\).

Step 3: Simplify the downstream equation

Simplify the downstream equation to express it in terms of speed: \(10 = (x+y)\frac{5}{4}\). Multiply both sides by \(\frac{4}{5}\) to get \(x+y = 8\).

Step 4: Set up the upstream equation

For upstream, the effective rate is the difference between the rowing rate in still water and the current rate: \(x - y\). They traveled the same 10 miles in 4 hours. So, the equation is \(10 = (x-y)4\).

Step 5: Simplify the upstream equation

Simplify the upstream equation to express it in terms of speed: \(10 = (x-y)4\). Divide both sides by 4 to get \(x-y = 2.5\).

Step 6: Solve the system of equations

Using the two equations: \(x+y = 8\) and \(x-y = 2.5\), add them to eliminate \(y\): \((x+y) + (x-y) = 8 + 2.5 \Rightarrow 2x = 10.5\). Divide by 2: \(x = 5.25\).

Step 7: Solve for the rate of the current

Substitute \(x = 5.25\) back into one of the original equations. Using \(x + y = 8\): \(5.25 + y = 8\), solve for \(y\): \(y = 8 - 5.25 = 2.75\).

Key Concepts

Solving Equations

Solving equations is a fundamental skill in mathematics and helps us find the values of unknown variables. In our problem about the Jonathan Schultz family canoeing down and up the Allegheny River, we used equations to identify the rates at which they were rowing in still water and the rate of the current. To solve an equation, you typically need to isolate the variable you are trying to find. In this exercise, we had two scenarios — downstream and upstream — which provided us with two equations.

• The downstream rate was expressed as \(x + y\), while the travel time of 10 miles equated to the equation \(10 = (x+y)\frac{5}{4}\).
• The upstream rate was \(x - y\), resulting in the equation \(10 = (x-y)4\).

By systematically simplifying these equations, you isolate variables, allowing you to solve for the unknowns like the rate of the current.

Downstream and Upstream Problems

Problems involving downstream and upstream scenarios are common in physics and mathematics when dealing with currents. Here, it's necessary to understand the effective speed of an object in a current.

• Downstream: When traveling downstream, the current assists the motion, so the effective speed is the sum of the object's speed in still water and the speed of the current. In this context, it translates into the equation \(x + y\), where \(x\) is the rowing speed, and \(y\) is the current speed.
• Upstream: Conversely, when going upstream, the current opposes the motion, reducing the effective speed. This leads to the equation \(x - y\).

Recognizing these relationships allows you to set up equations that reflect real-world scenarios, using the different travel times to solve for the unknown rate of the current.

System of Equations

The notion of a system of equations arises when we solve multiple equations simultaneously to find common solutions. In our canoeing problem, we derived two equations based on the downstream and upstream scenarios.

• The first equation was derived from the downstream scenario: \(x + y = 8\).
• The second equation was based on the upstream scenario: \(x - y = 2.5\).

To solve this system, we can use the method of elimination or substitution. In this problem, we added the two equations to eliminate \(y\), finding \(2x = 10.5\). By solving for \(x\), we obtained the rowing speed, which was then used to find \(y\), the rate of the current, by substitution back into one of the equations.Using a system of equations not only solves complex problems involving multiple variables but also provides a strategic way to approach real-world challenges involving interconnected quantities.