How many units should Mr. David produce daily?
Explanation:
The cost function C(x) = 240 + bx + cx2 C(20) = 240 + 20b + 400c C(40) = 240 + 40b + 1600c C(60) = 240 + 60b + 3600c
By conditions,
2/3 × C(20) = C(40) – C(20) ⇒ C(40) = 5/3 × C(20) ⇒ 240 + 40b + 1600c = 400 + 100b/3 + 2000c/3 ⇒ 20b/3 + 2800c/3 = 160 ⇒ 20b + 2800c = 480 ... (1)
Also,
½ × C(40) = C(60) – C(40) ⇒ 3/2 × C(40) = C(60) ⇒ 360 + 60b + 2400c = 240 + 60b + 3600c ⇒ c = 1/10 ⇒ b = 10 ... (from 1)
Profit P(x) for x units is 30x – C(x) ⇒ P(x) = 30x – 240 – 10x – x2/10 = – x2/10 +20x – 240
This is a quadratic expression whose maximum value will occur at x = -(20/2 × -1/10) = 100 ∴ x = 100
Hence, option (b).
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