The value of a particular model of car depreciates over time. The function \(\mathrm{V(t) = -2,500t + 35,000}\) models the...
GMAT Algebra : (Alg) Questions
The value of a particular model of car depreciates over time. The function \(\mathrm{V(t) = -2,500t + 35,000}\) models the value, in dollars, of the car \(\mathrm{t}\) years after it was purchased. Which of the following statements is the best interpretation of the number 35,000 in this context?
- The car's value decreases by $35,000 each year.
- The car's initial value was $2,500.
- The car's initial value was $35,000.
- The car will be worth $35,000 after 2,500 years.
The car's value decreases by \(\$35,000\) each year.
The car's initial value was \(\$2,500\).
The car's initial value was \(\$35,000\).
The car will be worth \(\$35,000\) after 2,500 years.
1. TRANSLATE the function components
- Given function: \(\mathrm{V(t) = -2,500t + 35,000}\)
- This is in slope-intercept form: \(\mathrm{y = mx + b}\)
- Here: \(\mathrm{slope = -2,500}\), \(\mathrm{y-intercept = 35,000}\)
2. INFER what the y-intercept means in context
- The y-intercept occurs when the independent variable equals zero
- In this case, when \(\mathrm{t = 0}\) (at the time of purchase)
- \(\mathrm{V(0) = -2,500(0) + 35,000 = 35,000}\)
3. TRANSLATE this mathematical result back to real-world meaning
- Since \(\mathrm{V(0) = 35,000}\), this represents the car's value at \(\mathrm{t = 0}\)
- \(\mathrm{t = 0}\) means "at the time of purchase" or "initially"
- Therefore, 35,000 is the car's initial value
Answer: C
Why Students Usually Falter on This Problem
Most Common Error Path:
Poor TRANSLATE reasoning: Students confuse the coefficient with the constant term and mix up their meanings in the context.
They might think "35,000 is the bigger number, so it must be how much the car decreases each year" or "2,500 looks like a reasonable initial value for a car." This confusion about which number represents which aspect of the function leads them to select Choice A ($35,000 decrease per year) or Choice B ($2,500 initial value).
Second Most Common Error:
Weak INFER skill: Students don't connect the y-intercept concept to the real-world context of "initial value."
They might recognize that 35,000 is the y-intercept mathematically, but fail to understand that in a time-based function, when \(\mathrm{t = 0}\) represents the starting point. This leads to confusion and they might select Choice D by randomly combining the numbers, or abandon systematic thinking entirely.
The Bottom Line:
This problem tests whether students can move fluidly between mathematical notation and real-world interpretation. Success requires recognizing that in linear functions modeling real situations, the y-intercept almost always represents the starting value or initial condition.
The car's value decreases by \(\$35,000\) each year.
The car's initial value was \(\$2,500\).
The car's initial value was \(\$35,000\).
The car will be worth \(\$35,000\) after 2,500 years.