a The pizza parlor has two separate charges: a fixed charge and a variable charge. Since the variable charge is the same for each number of guests, the function is linear.
B
b Profit is the total amount made minus the amount it cost to throw the party.
C
c Use the equation from Part B.
A
a C(x)=7x-300, x={0,1,2,3...}
B
b P(x)=43x-300, x={0,1,2,3...}
C
c At least 10 guests
Practice makes perfect
a Jill has a business hosting parties at a pizza parlor. Notice that using a room at the pizza parlor has two separate fees.
Fixed fee:& $300
Variable fee:& $7 per guest
Since the cost of the party increases by the same amount per guest, we can describe this with a linear function where 300 is the function's constant and 7 is the slope.
C(x)=7x+300
Notice that the function is only defined for whole numbers, as we can not have a fraction of a person attending the party.
C(x)=7x+300, x={0,1,2,3...}
b Jill's profit is the difference between what she earns and what she pays the pizza parlor. We already established the cost in Part A as the following function.
C(x)=7x+300
The profit is the amount she charges the customers times the number of customers that attends her party. Since she charges the same amount for each customer, $ 50, we get a second linear function with a slope of 50 and a y-intercept of 0. Let's label this function R for revenue.
R(x)=50x
The profit is the revenue minus the cost. We can label this function P.
P(x)=R(x)-C(x)
Let's substitute the function for the revenue and the cost and subtract them to find Jill's profit.
As in Part A, the domain is restricted to whole numbers.
P(x)=43x-300, x={0,1,2,3...}
c From Part B,we know the function that describes the profit. We want this function to be greater than or equal to 100, since Jill wants to make at least $100 in profit for each party. With this information, we can write the following inequality.
100≤ 43x-300
If we solve this inequality we can determine the minimum number of guests Jill must have to make a profit.
