- Topic
- Optimization
- Difficulty
- Standard
- Suggested time
- 5 min
- Marks
- 3
Optimization practice question 8
A closed box with a square base is to be built with a budget of $36 for materials. The base costs $2 per square metre and the other five faces cost $1 per square metre. Find the largest volume the box can have.
Give the answer in cubic metres.
Name the side of the square base and the height. List all six faces with their prices: what does the $36 budget then let you eliminate?
Let the base be by and the height . Cost: . Use this to write in terms of .
Then becomes a function of alone: . Maximise it.
A number.
Theory UBC Math 100 §8CLP-1 §3.5Key results
Worked solution
Strategy
The cost equation is the constraint; use it to eliminate the height, so that volume is a function of one variable. Then find and justify its maximum.
Working
Base at $2, top at $1, four sides at $1 each.
Volume in one variable; the domain is where h > 0.
One critical point in the domain, and V is concave down there, so it is the maximum.
A 2 m × 2 m × 3 m box.
Answer
The answer is 12 cubic metres.
Check
The 2 × 2 × 3 m box costs 8 + 4 + 24 = $36, and on that budget a 1.9 m or 2.1 m base holds only about 11.96 or 11.95 m³.
Where this goes wrong
Charging the base at $1 like the other faces, or forgetting the top of a closed box. The cost function must reflect exactly the faces described.
Leans on
Surface area of a box, from Functions, graphs and inverses
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