Select the correct answer.
Robert wants to throw a party for his friends. He has booked a banquet hall where he plans to serve dinner. The hall costs $200, and dinner costs $30 per person. If the final bill is $p, which function gives the number of friends at the party?
[Hint: First find the function for the final bill as a function of the number of friends, $x$.]
A. $30(p-200)$
B. $\frac{p+200}{30}$
C. $30(p+200)$
D. $\frac{p-200}{30}
Answer (2)
Define x as the number of friends and express the total cost p as p = 200 + 30 x .
Subtract 200 from both sides: p − 200 = 30 x .
Divide both sides by 30 to solve for x : x = 30 p − 200 .
The function representing the number of friends is 30 p − 200 .
Explanation
Problem Analysis Let's analyze the problem. Robert is throwing a party and has to pay a fixed cost for the hall and a variable cost depending on the number of friends attending. We need to find a function that expresses the number of friends in terms of the total bill.
Expressing the Total Cost Let x be the number of friends Robert invites. The total cost p can be expressed as the sum of the fixed hall cost and the variable dinner cost per person. So, we have: p = 200 + 30 x
Isolating the Variable Term Now, we need to solve this equation for x to find the number of friends in terms of the total bill p . First, subtract 200 from both sides of the equation: p − 200 = 30 x
Solving for the Number of Friends Next, divide both sides by 30 to isolate x :
x = 30 p − 200
Identifying the Correct Option Therefore, the function that gives the number of friends at the party is: x = 30 p − 200 This corresponds to option D.
Examples
Imagine you're organizing a school event and need to figure out how many students can attend based on your budget. If the venue costs a fixed amount and each student's meal has a per-person cost, this problem helps you determine the maximum number of attendees you can afford. By understanding how to express the number of attendees as a function of the total budget, you can make informed decisions about event planning and resource allocation. This kind of problem is also useful in business for calculating production capacity based on fixed and variable costs.
The number of friends Robert can invite to his party, based on the total cost p , can be expressed with the function 30 p − 200 . Therefore, the correct answer is option D. This function allows us to calculate how many friends can attend based on the total bill after accounting for the fixed banquet hall cost.
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