The daily cost, in dollars, to produce
x
gallons of handmade ice cream is
C
(
x
)
=
x
+
11
. The price-demand function, in dollars per gallon, is
p
(
x
)
=
−
0.15
x
+
20.8
Find the daily profit function.
Find the maximum daily profit. any gallons of ice cream need to be sold each day to maximize the profit.
Find the price to charge per gallon to maximize profit
Answers
9514 1404 393
Answer:
f(x) = -0.15x^2 +19.8x -11 . . . profit function$642.40 maximum daily profit66 gallons sold$10.90 per gallonStep-by-step explanation:
Given:
daily cost function c(x) = x +11
price-demand function p(x) = -0.15x +20.8
Find:
daily profit function
maximum daily profit
quantity sold for maximum profit
price for maximum profit
Solution:
The revenue is the product of price and demand:
x·p(x) = -0.15x^2 +20.8x
The profit is the difference between revenue and cost:
f(x) = x·p(x) -c(x) = -0.15x^2 +20.8x -x -11
f(x) = -0.15x^2 +19.8x -11 . . . . . daily profit function
__
The maximum profit will be had a the vertex of the curve, found where ...
x = -(19.8)/(2(-0.15)) = 66
f(66) = (-0.15·66 +19.8)66 -11 = 9.9·66 -11 = 642.40
The maximum profit is $642.40, when 66 gallons of ice cream are sold.
__
The price that will result in demand of 66 gallons of ice cream is ...
p(66) = -0.15(66) +20.8 = 10.9
The price to charge per gallon to maximize profit is $10.90.
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