4. (09.02 HC) The function f(x) = −x2 + 44x − 384 Part B: Determine the x-intercepts. What do these values mean in the context of the problem? (5 points)

Question

4.

(09.02 HC)

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? (5 points)



Part B: Determine the x-intercepts. What do these values mean in the context of the problem? (5 points)

Answer 1

f(x)=-x²+44x-484+484-384=100-(x-22)².

Part A: The vertex is the maximum point of f(x), and this occurs when x=22, when f(x)=100. This means that the maximum profit is $100 when 22 donut combos are sold.

Part B: f(x)=100-(x-22)²=(10-(x-22))(10+(x-22)). So f(x)=(32-x)(x-12), and the x intercepts are 32 and 12. This means that there is break even (neither profit nor loss) when 32 or 12 donut combos are sold. Note that the maximum profit occurs at the average of these two quantities—(32+12)/2=44/2=22. The x intercepts are symmetrical, being 10 donut combos on each side of the axis of symmetry.