Worksheets · Calculus

Optimization worksheet

Write the thing you are making big or small in terms of one variable, using the constraint to get rid of the other, then find where its derivative is zero. The constraint is what turns two unknowns into one.

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Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Optimization

Date Period

Answer each question.

  1. Solve optimization problems

    A rectangular paddock is split into two equal halves by a fence parallel to one side. With 24 m of fence in total, what is the largest area the whole paddock can have?
  2. A rectangular sign has an area of 9 cm2. What is the smallest its perimeter can be?
  3. A rectangular run is split into two equal halves by a fence parallel to one side. With 120 m of fence in total, what is the largest area the whole run can have?
  4. A rectangular pen is split into three equal strips by two fences parallel to one side. With 120 m of fence in total, what is the largest area the whole pen can have?

  1. Find the closest point on a curve

    Find the points on y=25x closest to the origin.
  2. Find the point on y=x closest to (32,0).
  3. Find the points on y=x2 closest to (0,32).
  4. Find the point on y=−3x−2 closest to (7,7).

  1. Find the best dimensions for a can

    A closed cylindrical can must hold 128π cubic inches. Find the radius and height that use the least material.
  2. An open-top cylindrical can must hold 512π cubic inches. Find the radius and height that use the least material.
  3. A closed cylindrical can must hold 864π cubic inches. The top and bottom cost 2 times as much per square inch as the side. Find the radius and height that cost the least.
  4. A closed cylindrical can must hold 48π cubic inches. The top and bottom cost 3 times as much per square inch as the side. Find the radius and height that cost the least.

  1. Find the price that maximizes revenue

    A shop sells 450 T-shirts a month at $24 each. Each $1 added to the price sells 5 fewer. What price gives the most revenue?
  2. At $32 a ticket, 580 tickets sell. Each $1 increase loses 5 buyers. What price gives the most revenue?
  3. A gym has 600 members paying $20 a month. Each $1 increase loses 25 members. What price gives the most revenue?
  4. A tour boat carries 590 riders a week at $23 a ride. Each $1 increase loses 10 riders. What price gives the most revenue?

  1. Solve box and rope optimization problems

    An open box is made from a 7 in by 15 in sheet by cutting equal squares from the corners. What size square gives the largest volume, and what is that volume?
  2. Poles 16 ft and 4 ft tall stand 21 ft apart. A rope runs from the top of one to a stake in the ground between them and up to the top of the other. Where should the stake go to use the least rope, and how long is it?
  3. An open box is made from an 11 in by 35 in sheet by cutting equal squares from the corners. What size square gives the largest volume, and what is that volume?
  4. An open box is made from a 6 in by 16 in sheet by cutting equal squares from the corners. What size square gives the largest volume, and what is that volume?

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

OptimizationAnswer keyVersion 1

Date Period

Answer each question.

  1. Solve optimization problems

    A rectangular paddock is split into two equal halves by a fence parallel to one side. With 24 m of fence in total, what is the largest area the whole paddock can have?24 m2
  2. A rectangular sign has an area of 9 cm2. What is the smallest its perimeter can be?12 cm
  3. A rectangular run is split into two equal halves by a fence parallel to one side. With 120 m of fence in total, what is the largest area the whole run can have?600 m2
  4. A rectangular pen is split into three equal strips by two fences parallel to one side. With 120 m of fence in total, what is the largest area the whole pen can have?450 m2

  1. Find the closest point on a curve

    Find the points on y=25x closest to the origin.(−5,−5) and (5,5)
  2. Find the point on y=x closest to (32,0).(1,1)
  3. Find the points on y=x2 closest to (0,32).(−1,1) and (1,1)
  4. Find the point on y=−3x−2 closest to (7,7).(−2,4)

  1. Find the best dimensions for a can

    A closed cylindrical can must hold 128π cubic inches. Find the radius and height that use the least material.r=4, h=8 inches
  2. An open-top cylindrical can must hold 512π cubic inches. Find the radius and height that use the least material.r=8, h=8 inches
  3. A closed cylindrical can must hold 864π cubic inches. The top and bottom cost 2 times as much per square inch as the side. Find the radius and height that cost the least.r=6, h=24 inches
  4. A closed cylindrical can must hold 48π cubic inches. The top and bottom cost 3 times as much per square inch as the side. Find the radius and height that cost the least.r=2, h=12 inches

  1. Find the price that maximizes revenue

    A shop sells 450 T-shirts a month at $24 each. Each $1 added to the price sells 5 fewer. What price gives the most revenue?$57
  2. At $32 a ticket, 580 tickets sell. Each $1 increase loses 5 buyers. What price gives the most revenue?$74
  3. A gym has 600 members paying $20 a month. Each $1 increase loses 25 members. What price gives the most revenue?$22
  4. A tour boat carries 590 riders a week at $23 a ride. Each $1 increase loses 10 riders. What price gives the most revenue?$41

  1. Solve box and rope optimization problems

    An open box is made from a 7 in by 15 in sheet by cutting equal squares from the corners. What size square gives the largest volume, and what is that volume?32 in squares; 72 in3
  2. Poles 16 ft and 4 ft tall stand 21 ft apart. A rope runs from the top of one to a stake in the ground between them and up to the top of the other. Where should the stake go to use the least rope, and how long is it?845 ft from the 16 ft pole; 29 ft of rope
  3. An open box is made from an 11 in by 35 in sheet by cutting equal squares from the corners. What size square gives the largest volume, and what is that volume?52 in squares; 450 in3
  4. An open box is made from a 6 in by 16 in sheet by cutting equal squares from the corners. What size square gives the largest volume, and what is that volume?43 in squares; 160027 in3

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

A rectangle has a perimeter of 40 cm. What is the largest its area can be?

  1. If one side is x, the other is 20 minus x, so the area is x times 20 minus x.
  2. The derivative is 20 minus 2x, which is zero at x equals 10.
  3. So the rectangle is a 10 by 10 square and the area is 100 square centimeters.

The answer is 100 cm2.

Where students go wrong

Forgetting the constraint and differentiating something with two variables in it. There is only one thing to choose here; the constraint is what says what the second one has to be once the first is chosen.

Also called maximum and minimum word problems, applied maximum and minimum, largest area, minimum cost, box problem or fence problem.

What you can put on this worksheet

Optimization
Two numbers differ by 20. What is the smallest their product can be? → −100
The closest point on a curve
Find the point on y=3x−5 closest to (−3,−4). → (0,−5)
The can that uses the least material
An open-top cylindrical can must hold 125π cubic inches. Find the radius and height that use the least material. → r=5, h=5 inches
Revenue and price
A gym has 400 members paying $12 a month. Each $1 increase loses 10 members. What price gives the most revenue? → $26
A box from a rectangle, and two poles
An open box is made from a 15 in by 40 in sheet by cutting equal squares from the corners. What size square gives the largest volume, and what is that volume? → 103 in squares; 2500027 in3

Questions about these worksheets

Yes. Every optimization sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the optimization questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 5: optimization, the closest point on a curve, the can that uses the least material, revenue and price and a box from a rectangle, and two poles. Tick as many as you want and set how many of each, or let it spread them evenly.

Forgetting the constraint and differentiating something with two variables in it. There is only one thing to choose here; the constraint is what says what the second one has to be once the first is chosen.