Question

The function f(x) = −x2 + 44x − 384 models the daily profit, in dollars, a shop makes for selling donut combos, where x is the number of combos sold and f(x) is the amount of profit. Part A: Determine the vertex. What does this calculation mean in the context of the problem? Part B: Determine the x-intercepts. What do these values mean in the context of the problem?

Answer 1

A)
The vertex can be determined using the completing square method.
f(x) = -(x² - 44x + 384)
= -(x² - 44x + 384 + 22² - 22²)
= -(x - 22)² - 384 + 22²
= -(x - 22)² + 100
The vertex is (22 , 100) and this means that the maximum profit(due to negative sign of coefficient of x) is $100 and is achieved when 22 combos are sold
B) 0 = -(x - 22)² + 100
x - 22 = 10
x = 32
x - 22 = -10
x = 12
This means the shop gains no profit when it sells 32 or 12 combos, and this can be due to the production cost being equal to the money gained via sales.