AP Calculus AB Name___________________________________
Optimization HW
Solve each optimization problem.
1) A farmer wants to construct a rectangular pigpen using 200 ft of fencing. The pen will be
built next to an existing stone wall, so only three sides of fencing need to be constructed
to enclose the pen. What dimensions should the farmer use to construct the pen with the
largest possible area?
2) A company has started selling a new type of smartphone at the price of $110 − 0.05x
where x is the number of smartphones manufactured per day. The parts for each
smartphone cost $50 and the labor and overhead for running the plant cost $6000 per day.
How many smartphones should the company manufacture and sell per day to maximize
profit?
3) A supermarket employee wants to construct an open-top box from a 16 by 30 cm piece of
cardboard. To do this, the employee plans to cut out squares of equal size from the four
corners so the four sides can be bent upwards. What size should the squares be in order
to create a box with the largest possible volume?
4) Which point on the graph of y = x is closest to the point (5, 0)?
5) A geometry student wants to draw a rectangle inscribed in a semicircle of radius 3. If one
side must be on the semicircle's diameter, what is the area of the largest rectangle that the
student can draw?
-1- Worksheet by Kuta Software LLC
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6) A graphic designer is asked to create a movie poster with a 162 in² photo surrounded by a
4 in border at the top and bottom and a 2 in border on each side. What overall
dimensions for the poster should the designer choose to use the least amount of paper?
7) Engineers are designing a box-shaped aquarium with a square bottom and an open top.
The aquarium must hold 256 m³ of water. What dimensions should they use to create an
acceptable aquarium with the least amount of glass?
8) An architect is designing a composite window by attaching a semicircular window on top
of a rectangular window, so the diameter of the top window is equal to and aligned with
the width of the bottom window. If the architect wants the perimeter of the composite
window to be 20 m, what dimensions should the bottom window be in order to create the
composite window with the largest area?
9) Two vertical poles, one 9 m high and the other 18 m high, stand 36 m apart on a flat field.
A worker wants to support both poles by running rope from the ground to the top of each
post. If the worker wants to stake both ropes in the ground at the same point, where
should the stake be placed to use the least amount of rope?
-2- Worksheet by Kuta Software LLC
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Answers to Optimization HW
1) A = the area of the pigpen x = the length of the sides perpendicular to the stone wall
Function to maximize: A = x(200 − 2x) where 0 < x < 100
Dimensions of the pigpen: 50 ft (perpendicular to wall) by 100 ft (parallel to wall)
2) p = the profit per day x = the number of items manufactured per day
Function to maximize: p = x(110 − 0.05x) − (50x + 6000) where 0 ≤ x < ∞
Optimal number of smartphones to manufacture per day: 600
3) V = the volume of the box x = the length of the sides of the squares
Function to maximize: V = (30 − 2x)(16 − 2x) ⋅ x where 0 < x < 8
10
Sides of the squares: cm
3
4) (
9 3 2
,
2 2 ) 5) 9 6) 13 in wide by 26 in tall
7) 8 m by 8 m by 4 m tall 40 20
8) m (width) by m (height)
4+π 4+π
9) 12 m from the short pole (or 24 m from the long pole)
-3- Worksheet by Kuta Software LLC
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