Problem 41 - Prealgebra and Introductory Algebra

Question

To assess the damage done by a fire, a forest ranger traveled 1080 mi by jet and then an additional 180 mi by helicopter. The rate of the jet was four times the rate of the helicopter. The entire trip took \(5 \mathrm{h}\). Find the rate of the jet.

Step-by-Step Solution

Answer

The rate of the jet is 240 miles per hour.

Step 1: Understand the problem and assign variables

Let's denote the rate of the helicopter as \(h\), hence the rate of the jet will be \(4h\). We also know the total time \(T\) is 5 hours, the distance traveled by helicopter \(D_h\) is 180 miles, and the distance traveled by jet \(D_j\) is 1080 miles. Remember that \(D = R \cdot T\), where \(D\) is distance, \(R\) is rate, and \(T\) is time.

Step 2: Set up the equations

We can write the following two equations from what we know: \(D_h = h \cdot T_h\) and \(D_j = 4h \cdot T_j\). Because the total time is 5 hours, we can also write this equation: \(T_h + T_j = 5\).

Step 3: Solve for \(T_j\) then \(h\)

Let's subsitute to get \(T_j\) equation in terms of \(h\): \(D_j = 4h \cdot T_j\) gives us \(T_j = D_j / (4h)\). Substituting for \(T_j\) into the time sum equation \(T_h + D_j / (4h) = 5\). One could then solve this equation numerically for \(h\). However, since \(T_j + T_h = 5hrs\), we can also write the \(T_h\) equation in terms of \(h\): \(T_h + 180 / h = 5\). Solving for \(h\) gives \(h = 60\). Thus, the rate of the helicopter is 60 miles per hour.

Step 4: Find the rate of the jet

Since it's given that the rate of jet (\(j\)) is four times the rate of helicopter, from \(j = 4h\), we substitute \(h = 60\) to find \(j = 240\). So the rate of the jet is 240 miles per hour.

Key Concepts

Distance Formula

Understanding the Distance Formula is key for solving rate problems like the one in this exercise. The formula to remember is \( D = R \, \times \, T \), where \( D \) represents distance, \( R \) represents rate, and \( T \) represents time. In simpler terms, how fast you travel multiplied by how long you travel gives you the total distance covered.
In the exercise, we had two different trips: one by jet and one by helicopter. Each of these can be analyzed separately using the Distance Formula:

• For the helicopter, \( D_h = h \, \times \, T_h \), where 180 miles is traveled.
• For the jet, \( D_j = 4h \, \times \, T_j \), covering 1080 miles.

By understanding this relationship, we can set up equations that help break down the problem into simpler parts before solving for the unknown values.

Algebraic Equations

Algebraic equations are the backbone of solving this type of problem. Each piece of known information about the rate, distance, or total time can contribute to forming an equation.
Let's look at how we use equations here:

• The rate of the helicopter is represented by \( h \), and it's given that the rate of the jet is four times that, or \( 4h \).
• The total trip, lasting 5 hours, is expressed with the time equation \( T_h + T_j = 5 \).

By equating the two time formulas \( T_h = \frac{180}{h} \) and \( T_j = \frac{1080}{4h} \) and adding them up to equal the total time of 5 hours, we form a solvable equation. Solving these will give values of \( h \) (helicopter's rate) and \( j \) (jet's rate) respectively.

Problem Solving Steps

Effective problem solving in math involves breaking down the problem into manageable steps and systematically tackling them. Here's how you approach the exercise:
First, understand the problem and define your variables. We denoted the helicopter rate as \( h \), and used that to represent the jet rate as \( 4h \). Next, we set up the equations based on the Distance formula for both helicopter and jet.
After setting up, substitute values and rearrange equations to isolate one variable. This simplifies solving for unknowns. Specifically:

• Substitute \( T_j \) in terms of \( h \) in the time equation \( T_h + T_j = 5 \).
• Combine into a single equation and solve for \( h \), giving us the helicopter’s speed.

Finally, use the solution (helicopter's rate) to determine the jet's rate. This logical sequence helps ensure every part of the problem is addressed and solved accurately.