The function \(\mathrm{p = 175(0.8)^n + 50}\) gives the estimated remaining video playback time p, in minutes, on a tablet...
GMAT Advanced Math : (Adv_Math) Questions
The function \(\mathrm{p = 175(0.8)^n + 50}\) gives the estimated remaining video playback time \(\mathrm{p}\), in minutes, on a tablet \(\mathrm{n}\) hours after continuous use begins, where \(\mathrm{0 ≤ n ≤ 8}\). Which statement is the best interpretation of the ordered pair \(\mathrm{(n, p) = (2, 162)}\) in this context?
The tablet's remaining playback time decreases by 162 minutes each hour after continuous use begins.
The tablet's remaining playback time decreases by 2 minutes every 162 hours after continuous use begins.
2 hours after continuous use begins, the tablet's remaining playback time is estimated to be 162 minutes.
162 hours after continuous use begins, the tablet's remaining playback time is estimated to be 2 minutes.
1. TRANSLATE the given information
- Given: Function \(\mathrm{p = 175(0.8)^n + 50}\) where \(\mathrm{p}\) = remaining playback time (minutes), \(\mathrm{n}\) = hours after use begins
- Given ordered pair: \(\mathrm{(n, p) = (2, 162)}\)
- This means when \(\mathrm{n = 2}\), then \(\mathrm{p = 162}\)
2. INFER what we need to verify and interpret
- First, let's verify this point actually lies on our function
- Then interpret what these specific values mean in context
- The key insight: ordered pairs show input-output relationships, not rates
3. SIMPLIFY by substituting to verify
- Substitute \(\mathrm{n = 2}\) into the function:
\(\mathrm{p = 175(0.8)^2 + 50}\)
\(\mathrm{p = 175(0.64) + 50}\) (use calculator)
\(\mathrm{p = 112 + 50 = 162}\) ✓
4. TRANSLATE the mathematical result back to context
- \(\mathrm{n = 2}\) means "2 hours after continuous use begins"
- \(\mathrm{p = 162}\) means "162 minutes of remaining playback time"
- Combined: "2 hours after continuous use begins, the tablet's remaining playback time is estimated to be 162 minutes"
Answer: C
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret ordered pair values as rates of change rather than specific input-output values.
They see (2, 162) and think "decreases by 162 every 2 hours" or "decreases by 2 every 162 hours," treating these as slope-like rate information instead of recognizing them as coordinates showing a specific point in time and its corresponding playback time.
This may lead them to select Choice A (162 minutes decrease per hour) or Choice B (2 minutes every 162 hours).
Second Most Common Error:
Poor TRANSLATE reasoning: Students reverse the roles of the variables, forgetting which represents time and which represents remaining playback.
They might think \(\mathrm{n = 162}\) and \(\mathrm{p = 2}\), leading to "162 hours after use begins, 2 minutes remain" instead of correctly identifying that n represents hours elapsed and p represents minutes remaining.
This may lead them to select Choice D (162 hours after use, 2 minutes remaining).
The Bottom Line:
This problem challenges students to distinguish between function values at specific points versus rates of change, and to maintain clear variable identification when translating between mathematical and contextual representations.
The tablet's remaining playback time decreases by 162 minutes each hour after continuous use begins.
The tablet's remaining playback time decreases by 2 minutes every 162 hours after continuous use begins.
2 hours after continuous use begins, the tablet's remaining playback time is estimated to be 162 minutes.
162 hours after continuous use begins, the tablet's remaining playback time is estimated to be 2 minutes.