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Classic optimization: wire, window, fence practice problems

The classic optimization problems: an open box from a sheet, a Norman window, a can, a cut wire. 25 problems with worked solutions; in 25 of them every equation is proved by a computer algebra system.

4 worked problems

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

Problem 3.667

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A Norman window is a rectangle topped by a semicircle. If its perimeter is \( \displaystyle 8 \) ft, what radius \( \displaystyle x \) of the semicircle lets in the most light (the largest area), and what is that area?
  1. Width 2x, height h = (8 − 2x − πx)/2 from the perimeter; A(x) = 2x·h + πx²/2.
    Reviewed
  2. \[ \frac{d}{d x} \left(\frac{\pi x^{2}}{2} + 2 x \left(- \frac{\pi x}{2} - x + 4\right)\right) = - 4 x - \pi x + 8 \]
    Differentiate.✓ Proved
  3. \[ \left. - 4 x - \pi x + 8 \right|_{\substack{ x=\frac{8}{\pi + 4} }} = 0 \]
    The only critical point in the domain is x = 8/(pi + 4).✓ Proved
  4. \[ \left. \frac{d^{2}}{d x^{2}} \left(\frac{\pi x^{2}}{2} + 2 x \left(\frac{- \pi x}{2} - x + 4\right)\right) \right|_{\substack{ x=\frac{8}{\pi + 4} }} = -4 - \pi \]
    The second derivative is negative: a maximum.✓ Proved
  5. \[ \left. \frac{\pi x^{2}}{2} + 2 x \left(\frac{- \pi x}{2} - x + 4\right) \right|_{\substack{ x=\frac{8}{\pi + 4} }} = \frac{32}{\pi + 4} \]
    The area there.✓ Proved
Answer \( x = \frac{8}{\pi + 4},\ \text{area} = \frac{32}{\pi + 4} \approx 4.4808 \)

✓ 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 3.668

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A wire \( \displaystyle 10 \) m long is cut in two. One piece (length \( \displaystyle x \)) is bent into a square, the other into a circle. What \( \displaystyle x \) makes the total area as small as possible, and what is that area?
  1. Square side x/4; circle circumference 10 − x, radius (10 − x)/(2π). A(x) = (x/4)² + (10 − x)²/(4π).
    Reviewed
  2. \[ \frac{d}{d x} \left(\frac{x^{2}}{16} + \frac{\left(10 - x\right)^{2}}{4 \pi}\right) = \frac{\pi x + 4 x - 40}{8 \pi} \]
    Differentiate.✓ Proved
  3. \[ \left. \frac{\pi x + 4 x - 40}{8 \pi} \right|_{\substack{ x=\frac{40}{\pi + 4} }} = 0 \]
    The only critical point in the domain is x = 40/(pi + 4).✓ Proved
  4. \[ \left. \frac{d^{2}}{d x^{2}} \left(\frac{x^{2}}{16} + \frac{\left(10 - x\right)^{2}}{4 \pi}\right) \right|_{\substack{ x=\frac{40}{\pi + 4} }} = \frac{\pi + 4}{8 \pi} \]
    The second derivative is positive: a minimum.✓ Proved
  5. \[ \left. \frac{x^{2}}{16} + \frac{\left(10 - x\right)^{2}}{4 \pi} \right|_{\substack{ x=\frac{40}{\pi + 4} }} = \frac{25}{\pi + 4} \]
    The total area there.✓ Proved
Answer \( x = \frac{40}{\pi + 4},\ \text{total area} = \frac{25}{\pi + 4} \approx 3.5006 \)

✓ 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 3.673

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A closed cylindrical can must hold \( \displaystyle 16 \pi \) cubic inches. What radius \( \displaystyle x \) uses the least material, and what is that surface area?
  1. h = 16*pi/(πx²) from the volume, so S(x) = 2πx² + 2πx·h = 2πx² + 32π/x.
    Reviewed
  2. \[ \frac{d}{d x} \left(2 \pi x^{2} + \frac{32 \pi}{x}\right) = \frac{4 \pi \left(x^{3} - 8\right)}{x^{2}} \]
    Differentiate.✓ Proved
  3. \[ \left. \frac{4 \pi \left(x^{3} - 8\right)}{x^{2}} \right|_{\substack{ x=2 }} = 0 \]
    The only critical point in the domain is x = 2.✓ Proved
  4. \[ \left. \frac{d^{2}}{d x^{2}} \left(2 \pi x^{2} + \frac{32 \pi}{x}\right) \right|_{\substack{ x=2 }} = 12 \pi \]
    The second derivative is positive: a minimum.✓ Proved
  5. \[ \left. 2 \pi x^{2} + \frac{32 \pi}{x} \right|_{\substack{ x=2 }} = 24 \pi \]
    The surface area there.✓ Proved
Answer \( x = 2,\ \text{surface area} = 24 \pi \approx 75.3982 \)

✓ 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 3.674

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An open box is made from a \( \displaystyle 36 \times 36 \) sheet by cutting equal squares of side \( \displaystyle x \) from the corners and folding up the sides. What \( \displaystyle x \) gives the largest volume, and what is it?
  1. V(x) = x(36 − 2x)², for 0 < x < 18.
    Reviewed
  2. \[ \frac{d}{d x} x \left(36 - 2 x\right)^{2} = \left(x - 6\right) \left(12 x - 216\right) \]
    Differentiate.✓ Proved
  3. \[ \left. 12 \left(x - 18\right) \left(x - 6\right) \right|_{\substack{ x=6 }} = 0 \]
    The only critical point in the domain is x = 6.✓ Proved
  4. \[ \left. \frac{d^{2}}{d x^{2}} x \left(36 - 2 x\right)^{2} \right|_{\substack{ x=6 }} = -144 \]
    The second derivative is negative: a maximum.✓ Proved
  5. \[ \left. x \left(36 - 2 x\right)^{2} \right|_{\substack{ x=6 }} = 3456 \]
    The volume there.✓ Proved
Answer \( x = 6,\ \text{volume} = 3456 \)

✓ 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 classic optimization: wire, window, fence practice

A closed cylindrical can must hold \( \displaystyle 432 \pi \) cubic inches. What radius \( \displaystyle x \) uses the least material, and what is that surface area?
Problem 3.665medium✓ Every equation proved
A rectangular pen of area \( \displaystyle 2400 \) m² is split in two by a fence parallel to two of its sides (\( \displaystyle x \) is the length of those sides). What \( \displaystyle x \) uses the least fencing, and how much is that?
Problem 3.666medium✓ Every equation proved
A wire \( \displaystyle 20 \) m long is cut in two. One piece (length \( \displaystyle x \)) is bent into a square, the other into a circle. What \( \displaystyle x \) makes the total area as small as possible, and what is that area?
Problem 3.669medium✓ Every equation proved
A Norman window is a rectangle topped by a semicircle. If its perimeter is \( \displaystyle 20 \) ft, what radius \( \displaystyle x \) of the semicircle lets in the most light (the largest area), and what is that area?
Problem 3.670medium✓ Nihil obstat
A wire \( \displaystyle 8 \) m long is cut in two. One piece (length \( \displaystyle x \)) is bent into a square, the other into a circle. What \( \displaystyle x \) makes the total area as small as possible, and what is that area?
Problem 3.671medium✓ Nihil obstat
A Norman window is a rectangle topped by a semicircle. If its perimeter is \( \displaystyle 16 \) ft, what radius \( \displaystyle x \) of the semicircle lets in the most light (the largest area), and what is that area?
Problem 3.672medium✓ Nihil obstat
A Norman window is a rectangle topped by a semicircle. If its perimeter is \( \displaystyle 10 \) ft, what radius \( \displaystyle x \) of the semicircle lets in the most light (the largest area), and what is that area?
Problem 3.675medium✓ Nihil obstat
A closed cylindrical can must hold \( \displaystyle 250 \pi \) cubic inches. What radius \( \displaystyle x \) uses the least material, and what is that surface area?
Problem 3.676medium✓ Every equation proved
An open box is made from a \( \displaystyle 30 \times 30 \) sheet by cutting equal squares of side \( \displaystyle x \) from the corners and folding up the sides. What \( \displaystyle x \) gives the largest volume, and what is it?
Problem 3.677medium✓ Nihil obstat
A rectangular pen of area \( \displaystyle 600 \) m² is split in two by a fence parallel to two of its sides (\( \displaystyle x \) is the length of those sides). What \( \displaystyle x \) uses the least fencing, and how much is that?
Problem 3.678medium✓ Every equation proved
A rectangular pen of area \( \displaystyle 1200 \) m² is split in two by a fence parallel to two of its sides (\( \displaystyle x \) is the length of those sides). What \( \displaystyle x \) uses the least fencing, and how much is that?
Problem 3.679medium✓ Every equation proved
An open box is made from a \( \displaystyle 12 \times 12 \) sheet by cutting equal squares of side \( \displaystyle x \) from the corners and folding up the sides. What \( \displaystyle x \) gives the largest volume, and what is it?
Problem 3.690medium✓ Every equation proved
A Norman window is a rectangle topped by a semicircle. If its perimeter is \( \displaystyle 12 \) ft, what radius \( \displaystyle x \) of the semicircle lets in the most light (the largest area), and what is that area?
Problem 3.691medium✓ Nihil obstat
An open box is made from a \( \displaystyle 6 \times 6 \) sheet by cutting equal squares of side \( \displaystyle x \) from the corners and folding up the sides. What \( \displaystyle x \) gives the largest volume, and what is it?
Problem 3.692medium✓ Nihil obstat
An open box is made from a \( \displaystyle 24 \times 24 \) sheet by cutting equal squares of side \( \displaystyle x \) from the corners and folding up the sides. What \( \displaystyle x \) gives the largest volume, and what is it?
Problem 3.693medium✓ Nihil obstat
A rectangular pen of area \( \displaystyle 5400 \) m² is split in two by a fence parallel to two of its sides (\( \displaystyle x \) is the length of those sides). What \( \displaystyle x \) uses the least fencing, and how much is that?
Problem 3.694medium✓ Every equation proved
A wire \( \displaystyle 12 \) m long is cut in two. One piece (length \( \displaystyle x \)) is bent into a square, the other into a circle. What \( \displaystyle x \) makes the total area as small as possible, and what is that area?
Problem 3.695medium✓ Nihil obstat
A wire \( \displaystyle 4 \) m long is cut in two. One piece (length \( \displaystyle x \)) is bent into a square, the other into a circle. What \( \displaystyle x \) makes the total area as small as possible, and what is that area?
Problem 3.696medium✓ Nihil obstat
A closed cylindrical can must hold \( \displaystyle 54 \pi \) cubic inches. What radius \( \displaystyle x \) uses the least material, and what is that surface area?
Problem 3.697medium✓ Every equation proved
An open box is made from a \( \displaystyle 18 \times 18 \) sheet by cutting equal squares of side \( \displaystyle x \) from the corners and folding up the sides. What \( \displaystyle x \) gives the largest volume, and what is it?
Problem 3.698medium✓ Nihil obstat
A closed cylindrical can must hold \( \displaystyle 128 \pi \) cubic inches. What radius \( \displaystyle x \) uses the least material, and what is that surface area?
Problem 3.699medium✓ Every equation proved