Question

The function below models a certain hotel's expected monthly profit, in thousands of dollars, when the price is
x
dollars per room per night.



f
(
x
)
=
−
x
2
+
200
x
−
8
,
400



Which form of the function shows the amount, in dollars, the hotel should charge per room per night to maximize its expected monthly profit?


f
(
x
)
=
−
(
x
−
100
)
2
+
1
,
600


f
(
x
)
=
−
(
x
+
140
)
(
x
+
60
)


f
(
x
)
=
(
x
−
140
)
(
−
x
+
60
)


f
(
x
)
=
(
x
−
100
)
2
−
1
,
600

Answer 1

Complete the square:
f(x) = −x^2 + 200x − 8,400
= −(x^2 − 200x) − 8,400
= −(x − 100)^2 + 1,600.

So the first option, f(x) = −(x − 100)^2 + 1,600, is correct. The profit is maximized at x = $100 per room per night.