The equation \(\mathrm{M = 2.4e^{0.35(t - 18)}}\) models the number M, in billions, of global mobile app downloads in the...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{M = 2.4e^{0.35(t - 18)}}\)
• M = number of downloads in billions
• t = years after 2000
• Need to determine what 2.4 represents

2. INFER the key strategy

• In exponential functions of the form \(\mathrm{y = ae^{bx}}\), the coefficient 'a' represents the y-value when the exponential part equals 1
• This happens when the exponent equals 0, since \(\mathrm{e^0 = 1}\)
• Strategy: Find when \(\mathrm{0.35(t - 18) = 0}\)

3. SIMPLIFY to find the critical point

• Set the exponent equal to zero: \(\mathrm{0.35(t - 18) = 0}\)
• Divide by 0.35: \(\mathrm{t - 18 = 0}\)
• Solve: \(\mathrm{t = 18}\)

4. TRANSLATE the time value to real context

• Since t represents years after 2000, \(\mathrm{t = 18}\) means the year 2018
• When \(\mathrm{t = 18}\):
  \(\mathrm{M = 2.4e^{0.35(18-18)}}\)
  \(\mathrm{= 2.4e^0}\)
  \(\mathrm{= 2.4(1)}\)
  \(\mathrm{= 2.4}\)

5. INFER the final interpretation

• 2.4 is the value of M when \(\mathrm{t = 18}\)
• Therefore, 2.4 represents the estimated number of downloads (in billions) in 2018

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that to interpret the coefficient in an exponential function, they need to find where the exponential term equals 1. Instead, they might try to plug in \(\mathrm{t = 0}\) (representing year 2000) or get confused about what each part of the equation represents.

Without this strategic insight, they might select Choice A thinking 2.4 represents the value in 2000, or get confused and guess randomly between the choices.

Second Most Common Error:

Conceptual confusion about exponential functions: Students might confuse the coefficient (2.4) with the growth rate (0.35) or think the coefficient represents some kind of yearly increase amount.

This leads them to select Choice C (thinking 2.4 is the percent increase) or Choice D (thinking 2.4 is a fixed yearly increase), not recognizing this is exponential, not linear growth.

The Bottom Line:

This problem requires understanding that in exponential functions, the coefficient represents the function value when the exponential part equals 1, which provides the reference point for the model.