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Optimization practice problems

Fences, boxes and numbers: set up a function, then find its maximum or minimum. 24 problems with worked solutions; in 24 of them every equation is proved by a computer algebra system.

Two nonnegative numbers add up to 16. What is the largest possible value of their product?

Problem 3.76medium✓ Every equation proved

A rectangular field along a straight river needs no fence on the river side. With 120 m of fencing, what is the largest area that can be enclosed?

Problem 3.77medium✓ Every equation proved

A rectangular field along a straight river needs no fence on the river side. With 100 m of fencing, what is the largest area that can be enclosed?

Problem 3.78medium✓ Every equation proved

Two nonnegative numbers add up to 12. What is the largest possible value of their product?

Problem 3.79medium✓ Every equation proved

Two nonnegative numbers add up to 10. What is the largest possible value of their product?

Problem 3.80medium✓ Every equation proved

A rectangular field along a straight river needs no fence on the river side. With 200 m of fencing, what is the largest area that can be enclosed?

Problem 3.81medium✓ Every equation proved

Equal squares are cut from the corners of a 18 in by 18 in sheet of cardboard and the sides folded up to make an open box. What cut maximizes the volume, and what is the volume?

Problem 3.82medium✓ Every equation proved

A farmer has 100 m of fencing to enclose a rectangular pen. What dimensions maximize the area, and what is the maximum area?

Problem 3.83medium✓ Every equation proved

A rectangular field along a straight river needs no fence on the river side. With 240 m of fencing, what is the largest area that can be enclosed?

Problem 3.84medium✓ Every equation proved

Two nonnegative numbers add up to 20. What is the largest possible value of their product?

Problem 3.85medium✓ Every equation proved

Equal squares are cut from the corners of a 10 in by 10 in sheet of cardboard and the sides folded up to make an open box. What cut maximizes the volume, and what is the volume?

Problem 3.86medium✓ Every equation proved

A farmer has 200 m of fencing to enclose a rectangular pen. What dimensions maximize the area, and what is the maximum area?

Problem 3.87medium✓ Every equation proved

Equal squares are cut from the corners of a 6 in by 6 in sheet of cardboard and the sides folded up to make an open box. What cut maximizes the volume, and what is the volume?

Problem 3.88medium✓ Every equation proved

A farmer has 120 m of fencing to enclose a rectangular pen. What dimensions maximize the area, and what is the maximum area?

Problem 3.89medium✓ Every equation proved

Equal squares are cut from the corners of a 8 in by 8 in sheet of cardboard and the sides folded up to make an open box. What cut maximizes the volume, and what is the volume?

Problem 3.90medium✓ Every equation proved

Equal squares are cut from the corners of a 12 in by 12 in sheet of cardboard and the sides folded up to make an open box. What cut maximizes the volume, and what is the volume?

Problem 3.166medium✓ Every equation proved

Two nonnegative numbers add up to 24. What is the largest possible value of their product?

Problem 3.167medium✓ Every equation proved

Two nonnegative numbers add up to 30. What is the largest possible value of their product?

Problem 3.168medium✓ Every equation proved

Equal squares are cut from the corners of a 24 in by 24 in sheet of cardboard and the sides folded up to make an open box. What cut maximizes the volume, and what is the volume?

Problem 3.169medium✓ Every equation proved

A farmer has 40 m of fencing to enclose a rectangular pen. What dimensions maximize the area, and what is the maximum area?

Problem 3.170medium✓ Every equation proved

Equal squares are cut from the corners of a 30 in by 30 in sheet of cardboard and the sides folded up to make an open box. What cut maximizes the volume, and what is the volume?

Problem 3.171medium✓ Every equation proved

A farmer has 80 m of fencing to enclose a rectangular pen. What dimensions maximize the area, and what is the maximum area?

Problem 3.172medium✓ Every equation proved

A farmer has 60 m of fencing to enclose a rectangular pen. What dimensions maximize the area, and what is the maximum area?

Problem 3.173medium✓ Every equation proved

A rectangular field along a straight river needs no fence on the river side. With 400 m of fencing, what is the largest area that can be enclosed?

Problem 3.174medium✓ Every equation proved