Interpret the following linear function model with the given context. Leslie wants to throw a birthday party at their local bowling alley. The space is $100 to rent with an additional charge of $15 per person attending. This scenario of total cost is modeled by the linear function: b (x) = 100 + 15x. Leslie gets a bill of $550 at the end of the evening. In addition to Leslie and Steve, how many other people attended this birthday party?

Question

GPT-4o mini · Answer

To interpret the linear function $ b(x) = 100 + 15x $, we can break down its components in the context of Leslie's birthday party: - The term $ 100 $ represents the fixed cost to rent the bowling alley. This is the base cost, regardless of how many people attend the party. - The term $ 15x $ represents the variable cost that depends on the number of attendees, where $ x $ is the number of people attending the party (excluding Leslie and Steve). The $ 15 $ indicates that there is an additional charge of $ $15 $ for each person attending. Now, Leslie received a bill of $ $550 $ at the end of the evening. We can set up the equation based on the total cost: $$ b(x) = 550 $$ Substituting the linear function into this equation gives: $$ 100 + 15x = 550 $$ Next, we can solve for $ x $: 1. Subtract $ 100 $ from both sides: $$ 15x = 550 - 100 $$ $$ 15x = 450 $$ 2. Now, divide both sides by $ 15 $: $$ x = \frac{450}{15} $$ $$ x = 30 $$ Thus, $ x = 30 $ means that there were 30 people attending the party in addition to Leslie and Steve. To find the total number of attendees, we need to add Leslie and Steve: $$ \text{Total attendees} = x + 2 = 30 + 2 = 32 $$ In summary, in addition to Leslie and Steve, **30 other people attended the birthday party**.