Math Practice

Optimization Practice Problems

Q: How should I type my answers?

Type a number, without units. Exact forms like sqrt(11)/2 are accepted, and so are decimals.

Write the quantity you want to maximize or minimize, use the constraint to get it in one variable, set the derivative to zero, and check that you have the absolute max or min. Read each question to see whether it wants the value or the dimension.

Questions

1.
Two positive numbers add up to 30. What is the largest possible value of their product?
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With , . at , and . The largest product is .
2.
A rectangle has area 64 m². What is the smallest possible perimeter, in meters?
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With width and length , . at , and for . The rectangle is an 8 by 8 square with perimeter 32.
3.
A company's profit is dollars for units. How many units maximize profit?
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at , and , so this is the absolute maximum.
4.
200 ft of fence encloses a rectangle against a barn wall, so only three sides need fence. What is the largest possible area, in ft²?
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With two sides of length and one of length , on . at . , and the endpoints give 0.
5.
An open box is made from an 18-by-18-inch sheet by cutting equal squares from the corners and folding up the sides. What is the largest possible volume, in in³?
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on . , which is 0 at and . , and the endpoints give 0.
6.
A closed box with a square base must hold 1,000 cm³. What is the smallest possible surface area, in cm²?
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With base side and height , . gives , so . , so the minimum is a 10 cm cube with .
7.
Find the shortest distance from the point to the curve . You can type sqrt.
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Minimize . , which is 0 at and . and at . The shortest distance is .
8.
A rectangle has its base on the x-axis and its top corners on . What is its largest possible area?
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on . at . , and the endpoints give 0.
9.
Positive numbers and satisfy . What is the largest possible value of ?
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With , on . at . , and the endpoints give 0.
10.
You're in a boat 4 km from the nearest point on a straight shore. A town is 10 km down the shore from that point. You row at 3 km/h and walk at 5 km/h. How many km down the shore should you land to reach town fastest?
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. gives , so and . h, less than and .
11.
A closed cylindrical can must hold cm³. What radius, in cm, uses the least material?
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, so . gives , so (and ). , so it's the minimum.
12.
All 100 apartments in a building rent at $800 a month. Each $20 increase in rent leaves one more apartment empty. What monthly rent, in dollars, maximizes revenue?
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After increases, . at , and . The rent is , with 70 apartments rented for a month.

Written by

Zohaib

MPhil in Mathematics, Founder of Mathovia

Published

22 October 2026

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