Question 1024480: The gym charges $180 for a yearly membership. There are currently 1000 members. For every $5 increase, the gym will lose 10 members. How much should the gym charge to maximuze its revenue?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! you know that 180 * 1000 equals the revenue, and that the revenue will be 180,000.
180 is the price that each member pays.
1000 is the number of members when the price is 180.
they tell you that each time you add 5 to the price, the number of members goes down by 10.
if you let x be the common arameter used, then you get:
(180 + 5x) * (1000 - 10x) = the revenue collected.
for each increase of 1 in the parameter of x, the price paid by each member goes up 5 and the number of members goes down 10.
when x = 0, the equation becomes 180 * 1000 = 180,000.
this is the same as we were given, so it looks good so far.
when x = 1, the equation becomes 185 * 990 = 183,150.
the price per member went up 5 dollars and the number of members went down by 10 and so the total revenue became equal to 183,150.
the following graph shows the relationship between y (the revenue) versus x (the increase in membership fees in increments of 5 and the drop in membership in increments of 10).
the graph shows that the maximum revenue is achieved when x = 32.
what happens when x = 32 is that the cost per member has gone from 180 to 180 + 5 * 32 = 340, and the membership has gone from 100 to 1000 - 10 * 32 = 680.
680 members paying 340 apiece brings in a revenue of 340 * 680 = 231,200.
that's shown on the graph.
the graph is the graph of a quadratic equation and is therefore a parabola.
the max/min point of a quadratic equation is given by the equation of x = -b/2a.
to determine the value of a and b, the equation has to be in standard form.
multiply (185 + 5x) * (1000 - 10x) and you will get the quadratic equation of -50x^2 + 3200x + 180000 after reordering the terms in descending order of degree.
set this equal to 0 and you get -50x^2 + 3200x + 180000 = 0
the equation is now in standard form.
a = -50
b = 3200
c = 180000
x = -b/2a becomes x = -3200 / -100 which becomes x = 32.
when x = 32, y becomes -50 * (3x)^2 + 3200 * 32 + 180000.
that gets you y = 231,200, as shown on the graph.
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