each $4 increase in price reduces their sales by 2 players.
find the maximum daily income from sales and the price they should charge to get it.
the total income each day is equal to 80 * 60 with the price being 80 and the quantity sold being 60.
let x equal the change in the price.
when the price goes up 4, the quantity goes down 2, therefore the change in the quantity is half the change in the price.
the equation to model this is:
total income = (80 + x) * (60 - x/2)
when x = 0, the total income is equal to 80 * 60 = 4800
when x = -4, the total income is equal to 76 * 62 = 4712
when x = 4, the total income is equal to 84 * 58 = 4872
this equation can be graphed by setting y = total income.
the equation becomes y = (80 + x) * (60 - x/2)
the first graph shows the maximum total income which we will derive algebraically further down.
the second graph shows the total income when the price goes down 4.
the third graph shows the total income when the price goes up 4.
the equation is y = (80 + x) * (60 - x/2)
perform the indicated multiplication and the equation becomes:
y = 4800 - 40 * x + 60 * x - x^2/2
combine like terms and rearrange the terms in descending order of degree to get:
y = -x^2/2 + 20 * x + 4800
x^2 can be shown at .5 * x^2, therefore the equation becomes:
y = -.5 * x^2 + 20 * x + 4800
this equation is in standard form of y = ax^2 + bx + c, where:
a = -.5
b = 20
c = 4800
the maximum value of this equation is when x = -b/2a.
-b/2a becomes -20 / (2 * -.5) which becomes -20 / -1 which becomes 20.
the equation is at its maximum value when x = 20.
when x = 20, y = -.5 * 20^2 + 20 * 20 + 4800 which results in y = 5000.
this means that total income, represented by y, is equal to a maximum value of 5000 when the price is 20 dollars above 80.
the price is 80 + 20 = 100 and the quantity sold is 60 - 10 = 50.
100 * 50 = 5000.
remember the equation.
total income = (80 + x) * (60 - x/2)
x represents the increase in dollars.
if the price goes up 4, then x = 4.
if the price does down 4, then x = -4.
for every increase of 4 dollars, the quantity goes down by 2.
for every decrease of 4 dollars, the quantity goes up 2.