Model constraints, maxima and velocity.
Example 1
A rectangle has perimeter 28m. What width maximises its area?
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Answer: 7
- Let width be w. Length =14−w.
- A(w)=14w−w2, so A′(w)=14−2w.
- Set A′=0:w=7. A′′=−2 confirms a maximum.
Example 2
Profit is P(x)=−6x2+96x−2, for 0≤x≤16. Find the quantity x that maximises profit.
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Answer: 8
- P′(x)=−12x+96.
- P′=0 gives x=8, which lies in the allowed interval.
- The quadratic opens down, so this is the maximum.
Example 3
A particle has displacement s(t)=t3−18t2 for t≥0. Find the positive time when it is at rest.
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Answer: 12
- Velocity v(t)=3t2−36t=3t(t−12).
- At rest means v=0. The roots are 0 and 12; the positive one is 12.