SOLUTION: Peter has 40 pumpkins to sell. He will sell them all if he sets his price at $1.00 each, but will sell one less pumpkin for each 5 cent increase in price. What price should he char

Question 66187: Peter has 40 pumpkins to sell. He will sell them all if he sets his price at $1.00 each, but will sell one less pumpkin for each 5 cent increase in price. What price should he charge to maximize his earnings?(Show algebraically)

thank you!
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
Peter has 40 pumpkins to sell. He will sell them all if he sets his price at $1.00 each, but will sell one less pumpkin for each 5 cent increase in price. What price should he charge to maximize his earnings?
:5
Then amt sold will be (40-x); the price will be (1+.05x), y = total earnings
:
y = pumpkins sold * price:
y = (40-x)(1+.05x)
FOIL
y = 40 + 2x - 1x - .05x^2
y = 40 + x - .05x^2
:
Put it in the standard y = ax^2 + bx + c form:
y = -.05x^2 + x + 40
:
The max revenue(y) will occur at the vertex. Vertex equation: x = -b/(2a)
In our equation a=-.05; b= 1
x = -1/(2*-.05)
x = -1/-.1
x = 10
That means that the cost = $1 + (10*.05) = $1.50 a piece for max earnings
Number of pumpkins sold will be 40 - 10 = 30 pumpkins sold at that price
Total revenue: 30 * 1.50 = $45
:
A graphical presentation will make it clear to you:

:
Notice that the max earnings occurs when x = 10
:
He only needs to sell 30, that leaves 10 that he can give away or make pies or something, right?
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