Question

C) the function f(x) = −x2 + 16x − 60 models the daily profit, in dollars, a shop makes for selling candles, where x is the number of candles 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

Given the function modeling the profit:
f(x)=-x^2+16x-60

a)a.determine the vertex. what does this calculation mean in the context of the problem?
The vertex form is given by y=(x-h)^2+k, where (h,k) is the vertex:
y=f(x)=-x^2+16x-60
y=x^2-16x+60
c=(-b/2a)^2
b=16
thus
c=(-16/2)^2=64
hence:
y=x^2-16x+64+60-64
y=(x-8)(x-8)-4
y=(x-8)^2-4
hence the vertex form will be:
y=(x-8)^2-4 the vertex is (8,-4)
The vertex represents the highest point of the graph which is the highest daily profits attained.

b] determine the x-intercepts. what do these values mean in the context of the problem?
let y=0 thus
0=−x2 + 16x − 60
factorizing the above we get:
0=x^2-16x+60
0=x^2-6x-10x+60
0=x(x-6)-10(x-6)
thus
x=6 and x=10
thus the x-intercepts are x=6 and x=10, they represent the breakeven point. The minimum number of units they can sell and not make any profit