Problem 48 - Algebra 1: Concepts and Skills

Question

A cheetah running 90 feet per second is 100 feet behind a gazelle running 70 feet per second. How long will it take the cheetah to catch up to the gazelle? Use the verbal model to write and solve a linear equation. Speed of cheetah \( \cdot \text { Time }=100+\text { Speed of gazelle } \cdot\text { Time }\)

Step-by-Step Solution

Answer

The cheetah will catch the gazelle after 5 seconds.

Step 1: Define the Variables

Let's denote the time it takes for the cheetah to catch up to the gazelle as \( t \) (in seconds). We are given the speed of the cheetah \( v_c = 90 \) feet/second and the speed of gazelle \( v_g = 70 \) feet/second. The initial distance is \( d = 100 \) feet.

Step 2: Write the Equation

We know that distance equals speed times time. So for the cheetah and the gazelle, their distances covered after time \( t \) would be equals to \( v_c \cdot t \) and \( v_g \cdot t \) respectively, plus the initial 100ft head start for the Gazelle. When the cheetah catches up to the gazelle, these distances are the same. This leads to the following equation: \( v_c \cdot t = d + v_g \cdot t \)

Step 3: Simplify the Equation

Substitute the known values into the equation: \( 90t = 100 + 70t \). Simplify the equation by bringing like terms together: \( 90t - 70t = 100 \). Thus, we get 20t = 100.

Step 4: Solve for the time t

Solving for \( t \) gives us \( t = \frac{100}{20} = 5 \) seconds

Step 5: Verify the solution

Substitute \( t = 5 \) back into the original equation to check the solution. LHS is \( 90 * 5 = 450 \) and RHS is \( 100 + 70 * 5 = 450 \). As the Left Hand Side (LHS) equals to the Right Hand Side (RHS), the answer is correct.

Key Concepts

Rate Problems

Rate problems are essential in understanding how different speeds affect motion over time. In essence, a rate problem typically involves determining the relationship between distance, speed, and time. The key to solving these problems is the formula: \( \text{Distance} = \text{Speed} \times \text{Time} \).
This formula helps us find out how far an object travels when it moves at a constant speed over a specific period of time. When working with rate problems, it's crucial to:

• Identify the known values: in the exercise, the speed of the cheetah and gazelle.
• Define the unknown, which in our case is the time \( t \).
• Understand the relationship and how changes in speed affect the time needed to cover a certain distance.

This understanding helps us construct a linear equation that accurately models the real-world scenario shown in the problem.

Algebraic Problem Solving

Algebraic problem solving involves forming equations based on given relationships and known quantities. It is an essential skill for breaking down complex problems into more manageable parts.
This approach includes defining variables, forming equations, and then solving these equations to find unknown quantities.Here's the concept breakdown:

• Defining Variables: Start by defining what each variable represents, just like we define \( t \) for time.
• Forming Equations: Use the given data to form linear equations. In this case, it was forming the equation \( 90t = 100 + 70t \) to represent the cheetah catching up with the gazelle.
• Solving Equations: Manipulate the formed equation to isolate the variable and solve for it. Here, we simplified \( 20t = 100 \) to find \( t = 5 \).

Each step requires careful consideration to ensure the equation accurately represents the scenario at hand.

Distance-Speed-Time Relationships

Understanding the distance-speed-time relationships forms the core of solving rate problems. The equation \( \text{Distance} = \text{Speed} \times \text{Time} \) not only allows us to find out one of these quantities if the other two are known but also helps in understanding how they interrelate.
Consider these relationships:

• Speed Increase or Decrease: An increase in speed while other factors remain constant will decrease the time required to cover a distance, explaining why the faster cheetah catches up to the gazelle.
• Distance Travelled:: If the time and speed are known, you can calculate the distance using the formula.
• Time Calculation:: Rearrange the formula to \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \) to solve for time if the distance and speed are known.

These relationships show the importance of the formula in predicting movement, crucial for solving various motion-related problems.