Paul charges people $25 to test the air quality in their homes. The device he uses to test air quality cost him $500.
a.) Write an equation that describes Paul's net profit as a function of the number of clients he gets.
b.) How many clients does he need to break-even (when f(x) = 0)
p(x) = 25x-500
a.) To calculate Paul's net profit as a function of the number of clients he gets, we need to consider his total revenue and total cost. His total revenue is simply the cost per client multiplied by the number of clients, which is given as $25 times the number of clients. Meanwhile, his total cost is the initial cost of the device he uses, which is $500.
Let's denote the number of clients as x. The equation that describes Paul's net profit (P) as a function of the number of clients he gets (x) can be written as:
P(x) = R(x) - C(x)
Where:
R(x) represents the total revenue, which is 25x (cost per client multiplied by the number of clients).
C(x) represents the total cost, which is a constant $500.
Thus, the equation that describes Paul's net profit is:
P(x) = 25x - 500
b.) To determine the number of clients Paul needs to break-even, we'll set his net profit equation (P(x)) equal to zero, because the break-even point occurs when the profit is zero.
P(x) = 25x - 500
0 = 25x - 500
To solve for x, we'll isolate the variable:
25x = 500
x = 500 / 25
x = 20
Therefore, Paul needs 20 clients to break-even, as indicated by f(x) equating to 0.