ACT (Enhanced)MathematicsHard
A project manager is analyzing the budget for a new initiative. The cost (C) of the project, in thousands of dollars, is given by the function C(t) = 50 + 10t + 0.5t², where t is the time in months from the project's start. What is the average rate of change of the project cost during the first 4 months?
- A$12,000 per month
- B$10,000 per month
- C$14,000 per month
- D$16,000 per month
Show answer & explanationAnswer & explanation
Correct answer: C. $14,000 per month
The average rate of change from t=a to t=b is given by (C(b) - C(a)) / (b - a). Here, a=0 and b=4. First, find C(0) = 50 + 10(0) + 0.5(0)² = 50. Next, find C(4) = 50 + 10(4) + 0.5(4)² = 50 + 40 + 0.5(16) = 50 + 40 + 8 = 98. The average rate of change is (C(4) - C(0)) / (4 - 0) = (98 - 50) / 4 = 48 / 4 = 12. Since the cost is in thousands of dollars, the rate is $12,000 per month.
Why the other options are wrong
- A. This is correct: (C(4) - C(0)) / (4 - 0) = (98 - 50) / 4 = 48 / 4 = 12. Since C is in thousands, it's $12,000.
- B. This might come from an error in calculating C(4) or the final division.
- D. This is a plausible distractor, perhaps from multiplying by 4 instead of dividing.
Average Rate of Change (Quadratic)
The average rate of change of a function over an interval [a, b] is the slope of the secant line connecting the points (a, f(a)) and (b, f(b)), calculated as (f(b) - f(a)) / (b - a).
- Formula: (f(b) - f(a)) / (b - a).
- Represents the 'average speed' or 'average slope'.
- Does not require calculus, just function evaluation.
Memory trick: Y-difference over X-difference, for an average slope.