Download section 3.5

3.5 Applications of Extrema
1) A very small company constructs and sells waterproof cell phone
cases. The cases sell for $20 each at the local flea market. The dollar
cost of producing x protectors is modeled by the following cost function
𝐶 (𝑥 ) = 𝑥 2 + 8𝑥 + 10 (I kept the numbers in the problem small to
make the math easier)
a) Create a revenue function
b) Create a profit function
c) Determine the number of cases the company should produce and
sell to achieve maximum profit.
d) What is the maximum profit?
2) 𝐶 (𝑥 ) = 𝑥 2 + 4𝑥 − 40 is the total daily cost to produce for x
bracelets. The bracelets sell for $8 each. (I kept the numbers in the
problem small to make the math easier)
a) Create a revenue function
b) Create a profit function
c) Determine the number of bracelets the company should produce
and sell to maximize profit.
d) What is the maximum profit?
3) Boxed greeting cards cost the distributor 60 cents/box. The revenue
generated from selling x boxes of greeting cards is modeled by the
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function: 𝑅(𝑥 ) = 120𝑥 − 𝑥 2 .
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a) Create a cost function
b) Create a profit function
c) Determine the number of boxes of greetings cards that should be
bought and sold to maximize profit.
d) What is the maximum profit?
4) A single widget cost the distributor $12.00 to produce. The revenue
generated from selling x widgets is modeled by the function: 𝑅 (𝑥 ) =
240𝑥 − 2𝑥 2 .
a) Create a cost function
b) Create a profit function
c) Determine the number of widgets that should be produced and sold
to maximize profit.
d) What is the maximum profit?
5) The marketing research department of Shank, a quarterly magazine
for beginning golfers, has determined that the price-demand equation
for the magazine is approximated by
𝑝 = 2.75 − 0.01𝑥
where x represents the number of magazines printed and sold each
quarter, in hundreds, and p is the price, in dollars, of the magazine. The
cost of printing, distributing, and advertising is given by
𝐶 (𝑥 ) = 0.003𝑥 2 + 0.5𝑥 + 5
a) Create a revenue function.
b) Create a profit function.
c) How many units must the company produce and sell to maximize
profit?
d) What is the maximum profit?
e) What price per unit must be charged to make maximum profit?
6) A headphone determines that in order to sell x units of a new
headphone, the price demand equation for the headphones is given by
p = 1000 − x. It also determines that the total cost of producing x units
is given by C(x) = 3000 − 20x .
a) Create a revenue function.
b) Create a profit function.
c) How many units must the company produce and sell to maximize
profit?
d) What is the maximum profit?
e) What price per unit must be charged to make maximum profit?
7) The daily production cost for a factory to manufacture x deluxe
contour chairs is given to be
𝐶 (𝑥 ) = 500 + 14𝑥 +
𝑥2
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. The price demand function is p = 150 − 𝑥 .
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a) Create a revenue function.
b) Create a profit function.
c) How many units must the company produce and sell to maximize
profit?
d) What is the maximum profit?
e) What price per unit must be charged to make maximum profit?
8) The Double D Corporation analyzed the production costs for one of
its products and determined that the daily cost function can be given by
𝐶 (𝑥 ) = 0.03𝑥 3 − 2.4𝑥 2 + 73.16𝑥 + 102.27
where x is the number of units produced each day. The price demand
function is given by:
𝑝 = 0.14𝑥 2 − 15.52𝑥 + 561.21
a) Create a revenue function.
b) Create a profit function.
c) How many units must the company produce and sell to maximize
profit?
d) What is the maximum profit?
e) What price per unit must be charged to make maximum profit?