- Topic
- Optimization
- Difficulty
- Standard
- Suggested time
- 5 min
- Marks
- 3
Optimization practice question 7
An open-topped box is made from a 30 cm by 30 cm square sheet of card by cutting a square of side x from each corner and folding up the sides. Which value of x gives the largest volume?
Where in can the largest volume occur? is times a squared bracket: which rules will its derivative need?
Differentiate using the product rule, treating as one factor.
After differentiating, factor out . One of the two roots lies outside the physical domain.
A number in centimetres.
Theory UBC Math 100 §8CLP-1 §3.5Key results
Worked solution
Strategy
Differentiate the volume function, factor rather than expand so the roots are visible, and discard any critical point outside the domain the physical situation allows.
Working
Product rule, with the chain rule on the squared bracket.
Factor out the common bracket instead of expanding. The roots are now readable.
x = 15 removes the base entirely and gives zero volume, so it is rejected.
The endpoints x → 0 and x → 15 both give volume 0, so x = 5 is the maximum.
Answer
The answer is x = 5 cm, giving a maximum volume of 2000 cm³.
Check
V(4) = 4 · 22² = 1936 and V(6) = 6 · 18² = 1944, both less than V(5) = 2000 cm³.
Where this goes wrong
Accepting x = 15 as a solution because it satisfies the derivative equation. At x = 15 the folded sides consume the whole sheet and the box has no base, so it is outside the domain.
Leans on
Domain from a physical constraint, from Functions, graphs and inverses
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