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?

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

D

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