Question

The function C=500+0.5xx−50
𝐶
=
500
+
0.5
𝑥
𝑥
−
50
gives the average cost of producing x widgets, where the startup cost is $500. Once the widgets are developed, they are manufactured for $0.50 each. The first 50 widgets will not be sold but will be produced as free samples.

What do the key features of the function reveal?

Answer 1

First rewrite the formula clearly. If x is the total number of widgets produced (the first 50 are free samples, so only x − 50 can be sold), the average cost per sold widget is

C(x) = (500 + 0.5x) / (x − 50).

Key features and what they mean

- Domain: x ≠ 50; for practical (positive sold) interpretation require x > 50.
Interpretation: you must produce more than the 50 free samples before you can sell any widgets.

- Vertical asymptote at x = 50.
Interpretation: as the number sold approaches 0 (i.e., you only produce the samples), the average cost per sold widget blows up to infinity because the fixed startup cost is spread over almost no sold units.

- Horizontal asymptote y = 0.5 (lim x→∞ C(x) = 0.5).
Interpretation: the lowest possible average cost per sold widget is $0.50, the variable manufacturing cost per widget; as you produce and sell many widgets the fixed $500 startup cost becomes negligible.

- Monotonicity: C′(x) = −525/(x − 50)^2 < 0 for x ≠ 50, so C is strictly decreasing on (50, ∞).
Interpretation: average cost per sold widget decreases as you sell (and produce) more widgets — you get economies of scale — but it never goes below $0.50.

- No meaningful x-intercept in the practical domain (no positive x where C=0).
Interpretation: average cost cannot be zero.

- Example values to see the effect:
- x = 51 → C = (500 + 0.5·51)/(1) ≈ $525.50 per sold widget (very large because only one sold unit absorbs the whole startup cost).
- x = 100 → C = 550/50 = $11 per sold widget.
- x = 1000 → C ≈ 1000/950 ≈ $1.05 per sold widget.
- As x grows further C → $0.50.

Bottom line: the fixed startup cost ($500) and the 50 free samples make average cost very large when few units are sold; producing and selling more units steadily reduces the average cost, approaching the per‑unit manufacturing cost of $0.50 but never going below it.