Lin fills a water tank at a constant rate. The function \(\mathrm{W(t) = 18t + 120}\) models the volume, in...
GMAT Algebra : (Alg) Questions
Lin fills a water tank at a constant rate. The function \(\mathrm{W(t) = 18t + 120}\) models the volume, in gallons, of water in the tank t minutes after she begins filling it. Which statement is the best interpretation of the y-intercept of the graph of \(\mathrm{y = W(t)}\) in the xy-plane in this context?
- Lin adds approximately 18 gallons of water each minute.
- The tank held approximately 18 gallons when she began filling it.
- The tank held approximately 120 gallons when she began filling it.
- The tank will be full after approximately 120 minutes.
Lin adds approximately 18 gallons of water each minute.
The tank held approximately 18 gallons when she began filling it.
The tank held approximately 120 gallons when she began filling it.
The tank will be full after approximately 120 minutes.
1. TRANSLATE the problem information
- Given information:
- \(\mathrm{W(t) = 18t + 120}\) models volume in gallons
- \(\mathrm{t}\) represents minutes after Lin begins filling
- Need to interpret the y-intercept
- What this tells us: We have a linear function in the form \(\mathrm{y = mx + b}\), where the y-intercept is the constant term.
2. INFER how to find the y-intercept
- The y-intercept occurs when the input variable equals zero
- We need to evaluate \(\mathrm{W(0)}\) to find this value
- This will tell us the volume at time \(\mathrm{t = 0}\) (when filling began)
3. SIMPLIFY to find the y-intercept value
- Calculate \(\mathrm{W(0)}\):
\(\mathrm{W(0) = 18(0) + 120}\)
\(\mathrm{= 0 + 120}\)
\(\mathrm{= 120}\) - The y-intercept is 120
4. TRANSLATE the mathematical result back to context
- \(\mathrm{W(0) = 120}\) means when \(\mathrm{t = 0}\) (at the start), there were 120 gallons
- Therefore: The tank held 120 gallons when Lin began filling it
Answer: C
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students confuse the slope (18) with the y-intercept (120) and their contextual meanings.
They see "18" in the function and think "18 gallons initially" because 18 is the first number they encounter. They don't properly connect that the y-intercept (the constant term) represents the starting value, while the coefficient of \(\mathrm{t}\) represents the rate of change.
This may lead them to select Choice B (18 gallons initially).
Second Most Common Error:
Poor TRANSLATE reasoning: Students correctly identify 120 as the y-intercept but misinterpret what this value represents in context.
They might think "120" relates to time rather than volume, not carefully reading that \(\mathrm{W(t)}\) outputs gallons, not minutes. They assume 120 must be a time measurement.
This may lead them to select Choice D (120 minutes to fill).
The Bottom Line:
This problem tests whether students can correctly connect mathematical components of linear functions to their real-world meanings. The key is recognizing that in \(\mathrm{y = mx + b}\), the b-value represents the starting amount when the input is zero.
Lin adds approximately 18 gallons of water each minute.
The tank held approximately 18 gallons when she began filling it.
The tank held approximately 120 gallons when she began filling it.
The tank will be full after approximately 120 minutes.