\(\mathrm{f(t) = 8{,}000(0.65)^t}\) The given function f models the number of coupons a company sent to their customers at the...
GMAT Advanced Math : (Adv_Math) Questions
\(\mathrm{f(t) = 8{,}000(0.65)^t}\)
The given function \(\mathrm{f}\) models the number of coupons a company sent to their customers at the end of each year, where \(\mathrm{t}\) represents the number of years since the end of 1998, and \(\mathrm{0 \leq t \leq 5}\). If \(\mathrm{y = f(t)}\) is graphed in the \(\mathrm{ty}\)-plane, which of the following is the best interpretation of the y-intercept of the graph in this context?
The minimum estimated number of coupons the company sent to their customers during the \(5\) years was \(1,428\).
The minimum estimated number of coupons the company sent to their customers during the \(5\) years was \(8,000\).
The estimated number of coupons the company sent to their customers at the end of \(1998\) was \(1,428\).
The estimated number of coupons the company sent to their customers at the end of \(1998\) was \(8,000\).
1. TRANSLATE the question requirements
- We need to interpret the y-intercept of \(\mathrm{f(t) = 8{,}000(0.65)^t}\)
- The y-intercept occurs where \(\mathrm{t = 0}\)
- We must understand what this means in the real-world context
2. SIMPLIFY to find the y-intercept value
- Substitute \(\mathrm{t = 0}\) into the function:
\(\mathrm{f(0) = 8{,}000(0.65)^0}\) - Since any number to the power of 0 equals 1:
\(\mathrm{f(0) = 8{,}000(1) = 8{,}000}\) - The y-intercept is the point \(\mathrm{(0, 8{,}000)}\)
3. TRANSLATE the contextual meaning
- t represents "years since the end of 1998"
- So \(\mathrm{t = 0}\) means "0 years since the end of 1998" = "at the end of 1998"
- \(\mathrm{f(0) = 8{,}000}\) means 8,000 coupons were sent at the end of 1998
4. INFER the correct answer choice
- Looking at the options, we need the choice that says:
- The number is 8,000 (not 1,428)
- It refers to the end of 1998 (not a minimum over 5 years)
- This matches Choice D exactly
Answer: D
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students recognize that the y-intercept occurs at \(\mathrm{t = 0}\) and correctly calculate \(\mathrm{f(0) = 8{,}000}\), but they misinterpret what "y-intercept" means in context. They might think the y-intercept represents the minimum or maximum value of the function over the given domain, rather than the specific value when \(\mathrm{t = 0}\).
This confusion about the contextual meaning of y-intercept leads them to consider Choices A or B, which mention "minimum estimated number" rather than the value at a specific time point.
Second Most Common Error:
Conceptual confusion about exponential functions: Some students might not remember that \(\mathrm{(0.65)^0 = 1}\), leading them to calculate \(\mathrm{f(0)}\) incorrectly. They might think \(\mathrm{(0.65)^0 = 0.65}\) or get confused about the zero exponent rule entirely.
This calculation error could lead them to select Choice C (1,428) if they somehow arrive at that incorrect value, or cause them to abandon systematic solution and guess.
The Bottom Line:
This problem tests whether students can bridge the gap between mathematical concepts (y-intercept) and real-world interpretation. The key challenge is recognizing that y-intercept specifically refers to the function value when the input is zero, not other characteristics like minimum or maximum values.
The minimum estimated number of coupons the company sent to their customers during the \(5\) years was \(1,428\).
The minimum estimated number of coupons the company sent to their customers during the \(5\) years was \(8,000\).
The estimated number of coupons the company sent to their customers at the end of \(1998\) was \(1,428\).
The estimated number of coupons the company sent to their customers at the end of \(1998\) was \(8,000\).