Digital SATMath: Advanced MathMedium
A production manager is analyzing the cost of manufacturing a new component. The total cost C(x) in dollars, for producing x units, is given by C(x) = 1000 + 50x + 0.1x^2. The company wants the average cost per unit to be no more than $70. Which inequality represents this situation?
- A(1000 + 50x + 0.1x^2) / x ≤ 70
- B70x ≤ 1000 + 50x + 0.1x^2
- C1000 + 50x + 0.1x^2 ≤ 70
- D1000 + 50x + 0.1x^2 ≥ 70x
Show answer & explanationAnswer & explanation
Correct answer: A. (1000 + 50x + 0.1x^2) / x ≤ 70
The average cost per unit is the total cost divided by the number of units, which is C(x) / x. The problem states that the average cost should be no more than $70, meaning it must be less than or equal to 70. Therefore, (1000 + 50x + 0.1x^2) / x ≤ 70.
Why the other options are wrong
- B. While mathematically equivalent to C, it is not the direct translation of 'average cost per unit to be no more than $70'.
- C. This incorrectly sets the total cost, not the average cost, to be no more than $70.
- D. This is an incorrect rearrangement of the inequality, or an attempt to compare total cost to 70x, which is not the average cost.
Average Cost Formula
The average cost per unit is calculated by dividing the total cost of production by the number of units produced.
- Average Cost (AC) = Total Cost (TC) / Quantity (Q).
- If TC is C(x) and Q is x, then AC = C(x) / x.
- Inequalities like 'no more than' translate to '≤'.
Memory trick: Total Cost Over Units, Less Than Limit!