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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.

Interactive: Optimization Explorer

3 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 3.77

medium
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?
  1. Let x be the two sides perpendicular to the river; the side parallel is 120 − 2x. Area A(x) = x(120 − 2x), 0 < x < 60.
    Reviewed
  2. \[ \frac{d}{d x} x \left(120 - 2 x\right) = 120 - 4 x \]
    A'(x).✓ Proved
  3. \[ \left. 120 - 4 x \right|_{\substack{ x=30 }} = 0 \]
    A'(x) = 0 at x = 30.✓ Proved
  4. \[ \frac{d^{2}}{d x^{2}} x \left(120 - 2 x\right) = -4 \]
    A'' < 0: a maximum.✓ Proved
  5. \[ \left. x \left(120 - 2 x\right) \right|_{\substack{ x=30 }} = 1800 \]
    30 m by 60 m.✓ Proved
Answer \( 1800 \)

✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 3.82

medium
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?
  1. Cutting squares of side x leaves a base 18 − 2x on a side and height x: V(x) = x(18 − 2x)², 0 < x < 9.
    Reviewed
  2. \[ \frac{d}{d x} x \left(18 - 2 x\right)^{2} = \left(x - 3\right) \left(12 x - 108\right) \]
    V'(x), factored.✓ Proved
  3. \[ \left. x \left(8 x - 72\right) + \left(18 - 2 x\right)^{2} \right|_{\substack{ x=3 }} = 0 \]
    V'(x) = 0 at x = 3 (the other root, x = 9, gives no box).✓ Proved
  4. \[ \left. x \left(18 - 2 x\right)^{2} \right|_{\substack{ x=3 }} = 432 \]
    The maximum volume.✓ Proved
  5. V is 0 at both ends of the interval and positive in between, so the interior critical point is the maximum.
    Reviewed
Answer \( 432 \)

✓ Nihil obstat Lines: 3 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 3.83

medium
A farmer has 100 m of fencing to enclose a rectangular pen. What dimensions maximize the area, and what is the maximum area?
  1. Let the sides be x and y. Then 2x + 2y = 100, so y = 50 − x, and the area is A(x) = x(50 − x) for 0 < x < 50.
    Reviewed
  2. \[ \frac{d}{d x} x \left(50.0 - x\right) = 50.0 - 2 x \]
    A'(x).✓ Proved
  3. \[ \left. 50.0 - 2 x \right|_{\substack{ x=25.0 }} = 0 \]
    A'(x) = 0 at x = 25.0000000000000.✓ Proved
  4. \[ \frac{d^{2}}{d x^{2}} x \left(50.0 - x\right) = -2 \]
    A'' < 0, so this is a maximum.✓ Proved
  5. \[ \left. x \left(50.0 - x\right) \right|_{\substack{ x=25.0 }} = 625.0 \]
    A square 25.0000000000000 m by 25.0000000000000 m.✓ Proved
Answer \( 625.0 \)

✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More optimization practice

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 100 m of fencing, what is the largest area that can be enclosed?
Problem 3.78medium✓ Nihil obstat
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✓ Nihil obstat
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✓ Nihil obstat
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✓ Nihil obstat
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✓ Nihil obstat
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✓ Nihil obstat
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✓ Nihil obstat
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✓ Nihil obstat
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✓ Nihil obstat
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✓ Nihil obstat
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✓ Nihil obstat
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✓ Nihil obstat
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✓ Nihil obstat
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✓ Nihil obstat
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✓ Nihil obstat