Problem 61 - Precalculus with Limits

Question

DVD Rentals The number of rentals of a newly released DVD of a horror film at a movie rental store decreased each week. At the same time, the number of rentals of a newly released DVD of a comedy film increased each week. Models that approximate the numbers \(N\) of DVDs rented are $$\left\\{\begin{array}{r}{N=360-24 x} \\ {N=24+18 x}\end{array}\right.$$

Step-by-Step Solution

Answer

The number of comedy and horror movie rentals will be the same in the 8th week after their release.

Step 1: Identifying the System of Equations

Identify the two given equations as a system of equations. System of equations often have a solution where all given equations are satisfied. In this case, the solution would represent the point in time when both films have been rented the same number of times. Here are the equations: 1) \(N_1=360-24x\), 2) \(N_2=24+18x\)

Step 2: Setting Equations Equal to Each Other

The number of rentals will be the same when \(N_1 = N_2\), this means we can set the two equations equal to one another and solve for \(x\). This gives us the equation \(360 - 24x = 24 + 18x\)

Step 3: Solving for x

Combine like terms on both sides of the equation to isolate \(x\). This provides: \(360 - 24 = 42x\) which simplifies to \(x = (360 - 24) / 42\)

Step 4: Evaluating x

Evaluate \(x\) by performing the calculation: \(x = (360 - 24) / 42 = 8\)

Key Concepts

Solving Equations

The process of solving equations is fundamental in mathematics, where you aim to find the value of unknown variables that satisfy given conditions. In the exercise about DVD rentals, we encounter a system of equations. By solving these equations, we determine when the number of horror and comedy DVDs rented will be equal.
This involves setting equations equal and solving for the variable. We start by having our equations:

• Horror Film Rentals: \(N_1 = 360 - 24x\)
• Comedy Film Rentals: \(N_2 = 24 + 18x\)

To find when both rental numbers are the same, set \(N_1\) equal to \(N_2\). Solve the resulting equation:

• \(360 - 24x = 24 + 18x\)
• Combine like terms: \(360 - 24 = 42x\)
• Calculate \(x\): \(x = \frac{336}{42} = 8\)

This demonstrates the systematic approach to isolating our variable and finding a value that satisfies both original equations.

Linear Equations

Linear equations are equations where the highest power of the variable is 1. They are fundamental for modeling relationships in mathematics and real-world scenarios. In the DVD rental case, both equations, \(N = 360 - 24x\) and \(N = 24 + 18x\), are linear.
The numbers 360, -24, 24, and 18 are coefficients that influence the slope and intercept of the graph of the equation. For example, the first equation:

• Intercepts at \(N = 360\) when \(x = 0\)
• Decreases rentals by 24 each week (slope = -24)

This linearity ensures that the total rentals decrease or increase steadily over time. Understanding the linear nature of these equations helps in predicting and visualizing trends, which is useful in business contexts like tracking sales or rentals over time.

Mathematical Modeling

Mathematical modeling involves using mathematical expressions to represent real-world scenarios accurately. The DVD rental problem uses models in the form of linear equations to simulate rental trends for horror and comedy films.
By using these models, one can predict future behavior. In this particular exercise, the store can see how DVDs' popularity evolves and plan their inventory and promotion strategies. Each component of the equation represents a part of the rental scenario:

• Initial number of rentals (360 for horror, 24 for comedy)
• Weekly change in rentals (-24 per week for horror, +18 per week for comedy)

Through modeling, businesses and researchers can make informed decisions based on a clear understanding of trends, using mathematics as a predictive tool.