Question

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

Answer 1

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Answer:

• f(x) = -0.15x^2 +19.8x -11 . . . profit function
• $642.40 maximum daily profit
• 66 gallons sold
• $10.90 per gallon

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.