SOLUTION: The operator of a hotel estimates that 500 rooms per night will be rented if the room per night is $100. For each $10 increase in the price of room, six fewer rooms per night be re

Question 1200565: The operator of a hotel estimates that 500 rooms per night will be rented if the room per night is $100. For each $10 increase in the price of room, six fewer rooms per night be rented. Determine a linear function that will predicted the number of rooms that will be rented per night for a given price room. Use this model to predict the number of rooms that will be rented if the room rate is $150/night.

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
i believe your simplified function will be y = -.6x + 560.

i derived it as follows:

start with two points.
the first point is (100,500).
the second point is (110,494).

from this, you can create a linear function of y = mx + b, where m is the slope and b is the y-intercept.

the formula for the slope is m = (y2 - y1) / (x2 - x1)

label the first point as (x1,y1) = (100,500).
label the second point as (x2,y2) = (110,494).

m = (y2 - y1) / (x2 - x1) m = (494 - 500) / (110 - 100).
simplify to get:
m = -6/10 = -.6

that's your slope.

when x = 100, you want y to be 500).
the point representing this becomes (100,500).
you can use this point to find the y-intercept.
replace y with 500 and x with 100 in the equation to get:
y = -.6x + b becomes:
500 = -.6 * 100 + b
simplify to get:
500 = -60 + b
add 60 to both sides of the equation to get:
560 = b
that's your y-intercept.

the equation becomes:
y = -.6x + 560

i graphed the eqwuation to show you that the equation is accurate.
the graph is shown below.

x is the price of the room for one night.
y is the number of rooms rented.

when x = 100, y = 500, which is as stated in the problem statement.
when x = 110, y = 494, which indicates that, as the price goes up 10 dollars, the number of rooms rented is 6 less.

this continues for all values of x and corresponding values of y.

the equation can be used to determine the number of rooms rented when the price is 150.
as seen on the graph, the number of rooms rented will be 470 when the price is 150.
the price went up by 5 * 10 = 50 and the number of rooms rented went down by 5 * 6 = 30.
you can derive this from the equation as follows:
the equation is y = -.6 * x + 560
when x = 150, the equation becomes y = -.6 * 150 + 560.
simplify to get y = 470.

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