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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Pick more than one for a sheet that mixes them.



Sigma Prep · sigmaprep.io/worksheets Name Optimization Date Period Answer each question. 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? A rectangular garden 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 garden can have? A rectangular paddock is split into two equal halves by a fence parallel to one side. With 96 m of fence in total, what is the largest area the whole paddock can have? A rectangular paddock is split into three equal strips by two fences parallel to one side. With 64 m of fence in total, what is the largest area the whole paddock can have? A rectangular garden is built against a long hedge, so only three sides need fencing. With 52 m of fence, what is the largest area the garden can have? A rectangular pen is built against a straight wall, so only three sides need fencing. With 60 m of fence, what is the largest area the pen can have? A rectangular sign has an area of 144 cm $^{2}$ . $e x t W h a t i s t h e s m a l l e s t i t s p e r i m e t e r c a n b e ?$ A rectangular paddock is built against a river, so only three sides need fencing. With 20 m of fence, what is the largest area the paddock can have? A rectangular garden is split into two equal halves by a fence parallel to one side. With 108 m of fence in total, what is the largest area the whole garden can have? A rectangular paddock is built against a river, so only three sides need fencing. With 72 m of fence, what is the largest area the paddock can have? A rectangular run is split into three equal strips by two fences parallel to one side. With 40 m of fence in total, what is the largest area the whole run can have? A rectangular window has an area of 144 cm $^{2}$ . $e x t W h a t i s t h e s m a l l e s t i t s p e r i m e t e r c a n b e ?$ A rectangular pen is built against a straight wall, so only three sides need fencing. With 80 m of fence, what is the largest area the pen can have? A rectangle has an area of 49 cm $^{2}$ . $e x t W h a t i s t h e s m a l l e s t i t s p e r i m e t e r c a n b e ?$ A rectangular paddock is built against a river, so only three sides need fencing. With 80 m of fence, what is the largest area the paddock can have? A rectangular pen is built against a straight wall, so only three sides need fencing. With 40 m of fence, what is the largest area the pen can have? A rectangular paddock is split into three equal strips by two fences parallel to one side. With 96 m of fence in total, what is the largest area the whole paddock can have? A rectangular window has an area of 121 cm $^{2}$ . $e x t W h a t i s t h e s m a l l e s t i t s p e r i m e t e r c a n b e ?$ A rectangular plot is split into three equal strips by two fences parallel to one side. With 96 m of fence in total, what is the largest area the whole plot can have? 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? Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Optimization

Date Period

Answer each question.

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?
A rectangular garden 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 garden can have?
A rectangular paddock is split into two equal halves by a fence parallel to one side. With 96 m of fence in total, what is the largest area the whole paddock can have?
A rectangular paddock is split into three equal strips by two fences parallel to one side. With 64 m of fence in total, what is the largest area the whole paddock can have?
A rectangular garden is built against a long hedge, so only three sides need fencing. With 52 m of fence, what is the largest area the garden can have?
A rectangular pen is built against a straight wall, so only three sides need fencing. With 60 m of fence, what is the largest area the pen can have?
A rectangular sign has an area of 144 cm $^{2}$ . $e x t W h a t i s t h e s m a l l e s t i t s p e r i m e t e r c a n b e ?$
A rectangular paddock is built against a river, so only three sides need fencing. With 20 m of fence, what is the largest area the paddock can have?
A rectangular garden is split into two equal halves by a fence parallel to one side. With 108 m of fence in total, what is the largest area the whole garden can have?
A rectangular paddock is built against a river, so only three sides need fencing. With 72 m of fence, what is the largest area the paddock can have?
A rectangular run is split into three equal strips by two fences parallel to one side. With 40 m of fence in total, what is the largest area the whole run can have?
A rectangular window has an area of 144 cm $^{2}$ . $e x t W h a t i s t h e s m a l l e s t i t s p e r i m e t e r c a n b e ?$
A rectangular pen is built against a straight wall, so only three sides need fencing. With 80 m of fence, what is the largest area the pen can have?
A rectangle has an area of 49 cm $^{2}$ . $e x t W h a t i s t h e s m a l l e s t i t s p e r i m e t e r c a n b e ?$
A rectangular paddock is built against a river, so only three sides need fencing. With 80 m of fence, what is the largest area the paddock can have?
A rectangular pen is built against a straight wall, so only three sides need fencing. With 40 m of fence, what is the largest area the pen can have?
A rectangular paddock is split into three equal strips by two fences parallel to one side. With 96 m of fence in total, what is the largest area the whole paddock can have?
A rectangular window has an area of 121 cm $^{2}$ . $e x t W h a t i s t h e s m a l l e s t i t s p e r i m e t e r c a n b e ?$
A rectangular plot is split into three equal strips by two fences parallel to one side. With 96 m of fence in total, what is the largest area the whole plot can have?
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?



Sigma Prep · sigmaprep.io/worksheets Name OptimizationAnswer keyVersion 1 Date Period Answer each question. 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? $24$ m $^{2}$ A rectangular garden 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 garden can have? $450$ m $^{2}$ A rectangular paddock is split into two equal halves by a fence parallel to one side. With 96 m of fence in total, what is the largest area the whole paddock can have? $384$ m $^{2}$ A rectangular paddock is split into three equal strips by two fences parallel to one side. With 64 m of fence in total, what is the largest area the whole paddock can have? $128$ m $^{2}$ A rectangular garden is built against a long hedge, so only three sides need fencing. With 52 m of fence, what is the largest area the garden can have? $338$ m $^{2}$ A rectangular pen is built against a straight wall, so only three sides need fencing. With 60 m of fence, what is the largest area the pen can have? $450$ m $^{2}$ A rectangular sign has an area of 144 cm $^{2}$ . $e x t W h a t i s t h e s m a l l e s t i t s p e r i m e t e r c a n b e ?$ $48$ cm A rectangular paddock is built against a river, so only three sides need fencing. With 20 m of fence, what is the largest area the paddock can have? $50$ m $^{2}$ A rectangular garden is split into two equal halves by a fence parallel to one side. With 108 m of fence in total, what is the largest area the whole garden can have? $486$ m $^{2}$ A rectangular paddock is built against a river, so only three sides need fencing. With 72 m of fence, what is the largest area the paddock can have? $648$ m $^{2}$ A rectangular run is split into three equal strips by two fences parallel to one side. With 40 m of fence in total, what is the largest area the whole run can have? $50$ m $^{2}$ A rectangular window has an area of 144 cm $^{2}$ . $e x t W h a t i s t h e s m a l l e s t i t s p e r i m e t e r c a n b e ?$ $48$ cm A rectangular pen is built against a straight wall, so only three sides need fencing. With 80 m of fence, what is the largest area the pen can have? $800$ m $^{2}$ A rectangle has an area of 49 cm $^{2}$ . $e x t W h a t i s t h e s m a l l e s t i t s p e r i m e t e r c a n b e ?$ $28$ cm A rectangular paddock is built against a river, so only three sides need fencing. With 80 m of fence, what is the largest area the paddock can have? $800$ m $^{2}$ A rectangular pen is built against a straight wall, so only three sides need fencing. With 40 m of fence, what is the largest area the pen can have? $200$ m $^{2}$ A rectangular paddock is split into three equal strips by two fences parallel to one side. With 96 m of fence in total, what is the largest area the whole paddock can have? $288$ m $^{2}$ A rectangular window has an area of 121 cm $^{2}$ . $e x t W h a t i s t h e s m a l l e s t i t s p e r i m e t e r c a n b e ?$ $44$ cm A rectangular plot is split into three equal strips by two fences parallel to one side. With 96 m of fence in total, what is the largest area the whole plot can have? $288$ m $^{2}$ 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? $450$ m $^{2}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

OptimizationAnswer keyVersion 1

Date Period

Answer each question.

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? $24$ m $^{2}$
A rectangular garden 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 garden can have? $450$ m $^{2}$
A rectangular paddock is split into two equal halves by a fence parallel to one side. With 96 m of fence in total, what is the largest area the whole paddock can have? $384$ m $^{2}$
A rectangular paddock is split into three equal strips by two fences parallel to one side. With 64 m of fence in total, what is the largest area the whole paddock can have? $128$ m $^{2}$
A rectangular garden is built against a long hedge, so only three sides need fencing. With 52 m of fence, what is the largest area the garden can have? $338$ m $^{2}$
A rectangular pen is built against a straight wall, so only three sides need fencing. With 60 m of fence, what is the largest area the pen can have? $450$ m $^{2}$
A rectangular sign has an area of 144 cm $^{2}$ . $e x t W h a t i s t h e s m a l l e s t i t s p e r i m e t e r c a n b e ?$ $48$ cm
A rectangular paddock is built against a river, so only three sides need fencing. With 20 m of fence, what is the largest area the paddock can have? $50$ m $^{2}$
A rectangular garden is split into two equal halves by a fence parallel to one side. With 108 m of fence in total, what is the largest area the whole garden can have? $486$ m $^{2}$
A rectangular paddock is built against a river, so only three sides need fencing. With 72 m of fence, what is the largest area the paddock can have? $648$ m $^{2}$
A rectangular run is split into three equal strips by two fences parallel to one side. With 40 m of fence in total, what is the largest area the whole run can have? $50$ m $^{2}$
A rectangular window has an area of 144 cm $^{2}$ . $e x t W h a t i s t h e s m a l l e s t i t s p e r i m e t e r c a n b e ?$ $48$ cm
A rectangular pen is built against a straight wall, so only three sides need fencing. With 80 m of fence, what is the largest area the pen can have? $800$ m $^{2}$
A rectangle has an area of 49 cm $^{2}$ . $e x t W h a t i s t h e s m a l l e s t i t s p e r i m e t e r c a n b e ?$ $28$ cm
A rectangular paddock is built against a river, so only three sides need fencing. With 80 m of fence, what is the largest area the paddock can have? $800$ m $^{2}$
A rectangular pen is built against a straight wall, so only three sides need fencing. With 40 m of fence, what is the largest area the pen can have? $200$ m $^{2}$
A rectangular paddock is split into three equal strips by two fences parallel to one side. With 96 m of fence in total, what is the largest area the whole paddock can have? $288$ m $^{2}$
A rectangular window has an area of 121 cm $^{2}$ . $e x t W h a t i s t h e s m a l l e s t i t s p e r i m e t e r c a n b e ?$ $44$ cm
A rectangular plot is split into three equal strips by two fences parallel to one side. With 96 m of fence in total, what is the largest area the whole plot can have? $288$ m $^{2}$
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? $450$ m $^{2}$

How to do these

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

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

The answer is $100$ cm $^{2}$ .

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.

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.

Optimization worksheet

Select your difficulty

Optimization

OptimizationAnswer keyVersion 1

How to do these

Where students go wrong

Questions about these worksheets

Are these optimization worksheets free?+

Do the optimization worksheets come with an answer key?+

Are these optimization worksheets printable?+

What grade are these optimization worksheets for?+

Can I get a different version of the same optimization worksheet?+

Is this a free alternative to Kuta Software for optimization?+

Can I choose how hard the optimization questions are?+

What do students get wrong with optimization?+