Select the correct answer.
A coffee shop begins the day with 75 bagels and sells an average of 10 bagels each hour. Function [tex]$b$[/tex] models the bagel inventory, [tex]$b(x), x$[/tex] hours after opening.
[tex]$b(x)=75-10 x$[/tex]
If the coffee shop wants to make a graph of function [tex]$b$[/tex], which values of [tex]$x$[/tex] should it include on the graph to include the relevant domain within the context?
A. [tex]$0 \leq x \leq 7.5$[/tex]
B. [tex]$0 \leq x \leq \infty$[/tex]
C. [tex]$-\infty \leq x \leq \infty$[/tex]
D. [tex]$0 \leq x \leq 75$[/tex]
Answer (1)
The number of hours x must be non-negative: x ≥ 0 .
The bagel inventory b ( x ) must be non-negative: b ( x ) ≥ 0 .
Solve the inequality 75 − 10 x ≥ 0 for x , which gives x ≤ 7.5 .
The relevant domain for x is 0 ≤ x ≤ 7.5 , so the answer is 0 ≤ x ≤ 7.5 .
Explanation
Understanding the Problem We are given the function b ( x ) = 75 − 10 x which models the bagel inventory at a coffee shop x hours after opening. We want to find the relevant domain for x in this context.
Non-negative Time First, the number of hours x must be non-negative, since we can't have negative time. So, x ≥ 0 .
Non-negative Bagels Second, the number of bagels b ( x ) must also be non-negative, since we can't have a negative number of bagels. So, b ( x ) ≥ 0 . This means 75 − 10 x ≥ 0 .
Solving the Inequality Now, let's solve the inequality 75 − 10 x ≥ 0 for x . Add 10 x to both sides to get 75 ≥ 10 x . Divide both sides by 10 to get 7.5 ≥ x , or x ≤ 7.5 .
Finding the Domain Combining the two conditions, we have x ≥ 0 and x ≤ 7.5 . Therefore, the relevant domain for x is 0 ≤ x ≤ 7.5 .
Final Answer The correct answer is A. 0 ≤ x ≤ 7.5 .
Examples
Let's say you're managing a small bakery. The problem we solved helps you understand how long your initial stock of baked goods will last, given a certain rate of sales. For instance, if you start with 75 croissants and sell 10 per hour, you now know that you'll run out after 7.5 hours. This helps you plan when to bake more, ensuring you don't run out and lose customers. Understanding the domain of the function allows you to make informed decisions about production and staffing.
