Box Office Receipts U.S. movie box office receipts, in...

Box Office Receipts U.S. movie box office receipts, in billions of dollars, are shown in 5-year increments from 1993 to 2013. (TABLE CAN'T COPY) These receipts $R$ are reasonably approximated by the linear model $$R=0.2844 x+5.535$$ where $x=0$ corresponds to $1993, x=5$ corresponds to $1998,$ and so on. Using the model, calculate the year in which the receipts first exceed each amount. $\begin{array}{lll}\text { (a) } \$ 7.6 \text { billion } & \text { (b) } \$ 10 \text { billion } & \end{array}$

Box Office Receipts U.S. movie box office receipts, in billions of dollars, are shown in 5-year increments from 1993 to 2013.
(TABLE CAN'T COPY)
These receipts $R$ are reasonably approximated by the linear model
$$R=0.2844 x+5.535$$
where $x=0$ corresponds to $1993, x=5$ corresponds to $1998,$ and so on. Using the model, calculate the year in which the receipts first exceed each amount.
$\begin{array}{lll}\text { (a) } \$ 7.6 \text { billion } & \text { (b) } \$ 10 \text { billion } & \end{array}$

Key Concepts

Linear Model

A linear model represents a relationship between two variables with a constant rate of change. In this context, the model relates movie box office receipts to time, allowing prediction of values by calculating using the formula y = mx + b.

Slope and Intercept

The slope (m) in a linear model represents the rate at which the dependent variable changes per unit change in the independent variable, while the intercept (b) is the value of the dependent variable when the independent variable is zero. They are key to understanding and interpreting the trend being modeled.

Solving Linear Equations and Inequalities

Solving linear equations involves isolating the variable to determine when a particular condition is met. This process is essential when finding the exact point at which a modeled value exceeds a given threshold.

Translating Model Variables to Real-World Context

When using a mathematical model to represent real-world scenarios, it's important to accurately translate the model's variables to actual time or other contextual measures. This step ensures that the model's predictions are correctly interpreted in terms of calendar years or other applicable units.

Transcript

00:01 So for problem 79, it's a multi -step problem.

00:05 We do have two different questions to answer.

00:09 But what we're really looking at is a table from 1993 to 2013.

00:17 So this is showing in billions of dollars the box office receipts, movie box office receipts.

00:26 So these receipts are outlined to us in a linear model, which is r equals 0 .244x plus 5 .535.

00:52 And then we're also told that the table shows us years of five -year increments.

00:59 So they also give us some clarification where x equals zero, that corresponds to 1993, x equals 5, corresponds to 1998, and so on.

01:11 So really, what x is is the number of years after 1993.

01:15 Zero would be zero years because 1993 is where we started.

01:19 One, 1994, and so on that way as well.

01:23 So part a asks us to find where our, what year where we first exceeded $7 .6 billion.

01:34 So i think that the text is very specific on how they word this because remember, we are talking about inequalities.

01:42 So they do want to know the year in which the receipts first exceeded each amount.

01:50 Now, if we were to talk about the solid year in and of itself, of course, we could round up to the nearest year.

01:59 And that's kind of what we're looking at in part a and b.

02:03 But since it did say where it first exceeded, it's possible that we're going to be looking at a of course where we've got more than one receipt because this is taken throughout the entire year that's what we have to assume so for part a we would set our problem up where it is greater than 7 .6 now all this is in billions of course so it have 0 .2844x plus 5 .535 where this is greater than 7 .6 okay to first exceed 7 .6 billion.

02:47 When we solve this, we would subtract and solve just like a normal inequality.

02:56 And then that would leave us with 2 .844x greater than 2 .065, and then divide both sides by 0 .244.

03:14 So x would be greater than about 7 .3.

03:20 Now i did round that.

03:21 To the nearest one decimal place.

03:22 But remember we're talking about years.

03:24 Okay, so 7 .3...

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