Extremal problems
Answers are for parents and teachers.
1
For a rectangle with sides x and 18 − x (m) find x that makes the area greatest and compute that area.
81
2
For the motion s(t) = −t³ + 15t² (m), when is the speed greatest and how large is it?
v = −3t² + 30t, a = −6t + 30 = 0 ⇒ t = 5 s; v(5) = −75 + 150 = 75 m/s.
3
The profit is P(x) = −4x² + 200x − 900 (thousand so‘m), where x is the number of items made in a month. How many items give the greatest profit, and how large is it?
P′ = −8x + 200 = 0 ⇒ x = 25; P(25) = −2500 + 5000 − 900 = 1600 thousand so‘m = 1,600,000 so‘m.
4
Among rectangles inscribed in a circle of radius 5 cm, find the dimensions and area of the one with the greatest area.
The sides are 2x and 2√(25 − x²), S = 4x√(25 − x²), 0 < x < 5. S′ = 4(25 − 2x²)/√(25 − x²) = 0 ⇒ x = 5/√2. The sides are 5√2 cm and 5√2 cm (a square), S = 50 cm².
5
A workshop makes Q(t) = −t³ + 9t² + 30t parts in t hours (0 ≤ t ≤ 8). Find the time when the productivity Q′(t) is greatest and that productivity.
Q′ = −3t² + 18t + 30, Q″ = −6t + 18 = 0 ⇒ t = 3 h (Q″ changes from + to −). Q′(3) = −27 + 54 + 30 = 57 parts per hour.