A company purchases a new machine for its manufacturing plant. The function \(\mathrm{V(t) = 80,000 - 5,000t}\) models the value,...
GMAT Advanced Math : (Adv_Math) Questions
A company purchases a new machine for its manufacturing plant. The function \(\mathrm{V(t) = 80,000 - 5,000t}\) models the value, in dollars, of the machine t years after its purchase, for \(\mathrm{0 \leq t \leq 10}\). Which of the following is the best interpretation of the number \(\mathrm{80,000}\) in this context?
- The yearly decrease in the value of the machine in dollars.
- The value of the machine in dollars after \(\mathrm{10}\) years.
- The initial value of the machine in dollars.
- The total depreciation of the machine in dollars over \(\mathrm{10}\) years.
1. TRANSLATE the function components
- Given: \(\mathrm{V(t) = 80,000 - 5,000t}\)
- \(\mathrm{V(t)}\) = value of machine in dollars
- \(\mathrm{t}\) = years after purchase
- Domain: \(\mathrm{0 \leq t \leq 10}\)
2. INFER what the question is asking
- We need to interpret what the number 80,000 represents in this real-world context
- This means finding what role 80,000 plays in the machine's value over time
3. INFER the meaning by examining \(\mathrm{t = 0}\)
- At the moment of purchase, \(\mathrm{t = 0}\)
- \(\mathrm{V(0) = 80,000 - 5,000(0) = 80,000}\)
- This shows 80,000 is the machine's value when first purchased
4. INFER by connecting to linear function structure
- This function has the form \(\mathrm{y = mx + b}\)
- Here: \(\mathrm{b = 80,000}\) (y-intercept) and \(\mathrm{m = -5,000}\) (slope)
- The y-intercept always represents the initial value when the input is zero
Answer: C. The initial value of the machine in dollars
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students focus on the more 'active' coefficient \(\mathrm{-5,000}\) rather than the constant term 80,000, thinking the coefficient must be the answer since it's connected to the variable \(\mathrm{t}\).
They might reason: 'The problem is about a machine losing value over time, so the important number must be the one connected to time.' This leads them to select Choice A (yearly decrease).
Second Most Common Error:
Poor TRANSLATE reasoning: Students calculate \(\mathrm{V(10) = 30,000}\) and think this final value is what 80,000 represents, confusing the meaning of the constant with calculated outputs.
This calculation-focused approach may lead them to select Choice B (value after 10 years).
The Bottom Line:
Success requires recognizing that in linear functions, the constant term (y-intercept) represents the starting value, not the most mathematically 'active' component.