- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
The definition of the derivative practice question 4
A company’s weekly cost of making bicycle helmets is a linear function of . A week in which it makes 200 helmets costs $5400, and a week in which it makes 350 helmets costs $7650. Find the company’s fixed cost: the weekly cost when no helmets are made.
Give the answer in dollars.
Two weeks give two points on the cost line. What must you find from them before you can follow the line back to no helmets?
The marginal cost is the slope of the cost line: the change in cost divided by the change in the number of helmets.
The fixed cost is the intercept . Start from one of the given points and follow the line back to .
A whole number of dollars.
Theory UBC Math 100 §3.1Key results
Worked solution
Strategy
A linear cost function is C(x) = mx + b. The slope m is the marginal cost, what one more helmet costs; the intercept b is the fixed cost, paid even in a week when nothing is made. Two points determine both.
Working
Slope from the two points: the marginal cost is $15 per helmet.
Substitute either point into C(x) = 15x + b to find the intercept.
Check with the other point.
Answer
The answer is the cost line is C(x) = 15x + 2400, so the fixed cost is 2400 dollars a week.
Check
Both weeks agree: 5400 − 200 × 15 and 7650 − 350 × 15 both equal 2400.
Where this goes wrong
Giving the marginal cost, $15 per helmet, instead of the fixed cost. The slope says what one more helmet costs; the fixed cost is where the line meets x = 0.
Leans on
Slope of a line through two points, from Functions, graphs and inverses
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