In 2010, a streaming platform had 40,000 subscribers. The number of subscribers increases by 18% every 3 years, on average,...

1. TRANSLATE the problem information

• Given information:
  • Initial subscribers in 2010: 40,000
  • Growth rate: 18% every 3 years (not every year!)
  • Growth pattern: exponential
  • \(\mathrm{t}\) = years after 2010

• What this tells us: We need an exponential model, but the growth doesn't happen annually.

2. INFER the model structure

• For exponential growth: \(\mathrm{S(t) = initial \times (growth~factor)^{(number~of~periods)}}\)
• The tricky part: Growth happens every 3 years, not every year
• If \(\mathrm{t}\) is measured in years, then the number of 3-year periods in \(\mathrm{t}\) years is \(\mathrm{t/3}\)
• Growth factor = \(\mathrm{1 + 0.18 = 1.18}\) per 3-year period

3. INFER the correct function

• Putting it together: \(\mathrm{S(t) = 40,000 \times (1.18)^{(t/3)}}\)
• This matches choice (B)

4. Verify the model works

• At \(\mathrm{t = 0}\): \(\mathrm{S(0) = 40,000 \times (1.18)^0 = 40,000}\) ✓
• At \(\mathrm{t = 3}\): \(\mathrm{S(3) = 40,000 \times (1.18)^1 = 47,200}\) (18% increase from 40,000) ✓

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students read "18% every 3 years" but mentally process it as "18% every year," completely missing the critical time period specification.

They incorrectly set up \(\mathrm{S(t) = 40,000(1.18)^t}\), thinking the growth factor applies annually.

This leads them to select Choice A (\(\mathrm{S(t) = 40,000(1.18)^t}\)).

Second Most Common Error:

Poor INFER reasoning: Students recognize that growth happens every 3 years but get confused about how to adjust the exponent. They might think "3 years means multiply by 3" instead of "divide by 3."

This backward reasoning leads them to \(\mathrm{S(t) = 40,000(1.18)^{(3t)}}\), which would mean 18% growth every 1/3 year.

This causes them to select Choice D (\(\mathrm{S(t) = 40,000(1.18)^{(3t)}}\)).

The Bottom Line:

This problem tests whether students can properly TRANSLATE time relationships in word problems and INFER how to adjust standard formulas when the growth period doesn't match the time variable units. The key insight is recognizing that \(\mathrm{t/3}\) converts years into 3-year periods.