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Chapter 8: Problem 75

The total revenue generated by a film comes from two sources: box-office ticket sales and the sale of merchandise associated with the film. It is estimated that for a very popular film such as Spiderman or Harry Potter, the revenue from the sale of merchandise is four times the revenue from ticket sales. Assume this is true for the film Spiderman, which grossed a total of \(\$ 3\) billion. Find the revenue from ticket sales and the revenue from the sale of merchandise.

Short Answer

Expert verified
The revenue from tickets sales is $0.6 billion and the revenue from merchandise sales is $2.4 billion.

Step by step solution

01

Set up the equations

We begin by setting up the two equations that can be formed based on the information given in the problem. Let T denote the ticket sales, and M denote the revenue from merchandise.\n1) \(M = 4T\)\n2) \(T + M = 3\) billion.
02

Substitute the first equation into the second

To solve, substitute M from the first equation into the second equation, replacing M with 4T. So the equation now becomes, \( T + 4T = 3\).
03

Solve the equation

Combine the alike terms on the left side of the equation gets \( 5T = 3\) billion dollars. To isolate T divide both sides of the equation by 5,\( T = \frac{3}{5} = 0.6\)billion dollars.
04

Find the revenue from merchandise sales

Now that we have T, we can find M by substituting T into the first equation. This gives us \( M = 4 \times 0.6 = 2.4 \) billion dollars.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equation Solving
When tackling a problem in revenue management, you'll often need to solve equations. That's because equations let us find unknown values using known relationships. In our exercise, we need to determine two financial figures related to a movie's revenue: ticket sales and merchandise sales. We start by setting up equations based on given conditions.

Here are the steps to solve such equations:
  • Identify the unknowns: In this case, the unknowns are the ticket revenue (\(T\)) and merchandise revenue (\(M\)).
  • Formulate the relationships: The exercise states that merchandise revenue is four times ticket revenue (\(M = 4T\)). Also, the total revenue is \(3\) billion dollars (\(T + M = 3\) billion).
  • Substitute known equations in each other: To solve for one variable, we substitute \(M\) (from \(M = 4T\)) into the total revenue equation, which yields \(T + 4T = 3\).
  • Solve the simplified equation: Combining terms gives \(5T = 3\). Finally, solve for \(T\) by dividing both sides by \(5\), getting \(T = 0.6\) billion dollars.
Equation solving helps us find the ticket sales revenue, which then allows us to calculate further financial outcomes.
Algebraic Manipulation
Algebraic manipulation is a powerful tool in mathematics, allowing us to rearrange and simplify equations to find solutions. It's especially useful in problems involving revenue management.

To understand how algebraic manipulation applies to our exercise, let's follow these steps:
  • Starting with the substitution: Once we substitute \(M = 4T\) into the total revenue equation, we simplify the situation to one equation with one unknown (\(T\)).
  • Combine like terms: The equation \(T + 4T = 3\) billion shows that \(T\) terms can be combined to give \(5T = 3\) billion.
  • Isolate the variable: The goal is to make \(T\) the subject of the formula. We do this by dividing all terms by \(5\), resulting in \(T = 0.6\) billion dollars.
  • Verify and use the solution: With \(T\) determined, substitute back into \(M = 4T\) to find \(M = 2.4\) billion dollars.
Through algebraic manipulation, we effectively rearrange and resolve complex relationships to achieve clear numerical results.
Mathematical Modeling
Mathematical modeling is the process of representing real-world scenarios with mathematical formulas. It's crucial in revenue management since it allows businesses to predict and optimize financial outcomes.

In our particular exercise, the situation is modeled using a set of equations, representing how different sources contribute to the total revenue of a film:
  • Define variables: Here, the ticket sales (\(T\)) and merchandise sales (\(M\)) are chosen as variables to build the model.
  • Understand the relationship: The relationship \(M = 4T\) models the assumption that merchandise sales are four times the ticket sales, which is typical for blockbuster films.
  • Modeling the total revenue: The linear equation \(T + M = 3\) billion encapsulates the entire revenue reached by the movie.
  • Solving the model: Using the mathematical structures, we solve for our variables, achieving a practical insight into the revenue distribution: ticket sales are \(0.6\) billion, and merchandise is \(2.4\) billion.
Mathematical models like this are indispensable for making data-driven decisions in business management and strategy.

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Most popular questions from this chapter

In this set of exercises, you will use the method of solving linear systems using matrices to study real-world problems. A grocery store carries two brands of diapers. For a certain week, the number of boxes of Brand \(A\) diapers sold was 4 more than the number of boxes of Brand B diapers sold. Brand A diapers cost \(\$ 10\) per box and Brand B diapers cost \(\$ 12\) per box. If the total revenue generated that week from the sale of diapers was \(\$ 172,\) how many of each brand did the store sell?

A golf club manufacturer makes a profit of \(\$ 3\) on a driver and a profit of \(\$ 2\) on a putter. To meet dealer demand, the company needs to produce between 20 and 50 drivers and between 30 and 50 putters each day. The maximum number of clubs produced each day by the company is 80. How many of each type of club should be produced to maximize profit?

A telephone company manufactures two different models of phones: Model 120 is cordless and Model 140 is not cordless. It takes 1 hour to manufacture the cordless model and 1 hour and 30 minutes to manufacture the traditional phone. At least 300 of the cordless models are to be produced. The manufacturer realizes a profit per phone of \(\$ 12\) for Model 120 and \(\$ 10\) for Model \(140 .\) If at most 1000 hours are to be allocated to the manufacture of the two models combined, how many of each model should be produced to maximize the total profit?

A manufacturer wants to make a can in the shape of a right circular cylinder with a volume of \(45 \pi\) cubic inches and a lateral surface area of \(30 \pi\) square inches. The lateral surface area includes only the area of the curved surface of the can, not the area of the flat (top and bottom) surfaces. Find the radius and height of the can.

Time Bill can't afford to spend more than \(\$ 90\) per month on transportation to and from work. The bus fare is only \(\$ 1.50\) one way, but it takes Bill 1 hour and 15 minutes to get to work by bus. If he drives the 20 -mile round trip, his one-way commuting time is reduced to 1 hour, but it costs him S.45 per mile. If he works at least 20 days per month, how often does he need to drive in order to minimize his commuting time and keep within his monthly budget?
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