According to Moore's law, the number of transistors included on microprocessors doubles every 2 years. In 1985, a microprocessor was...

1. TRANSLATE the problem information

• Given information:
  • Starting point: 275,000 transistors in 1985
  • Growth pattern: doubles every 2 years
  • Target: \(\mathrm{1.1~million = 1,100,000}\) transistors

• What this tells us: We need an exponential growth model where the quantity doubles every 2-year period.

2. INFER the mathematical approach

• Since the quantity doubles every 2 years, we need to track how many 2-year periods occur in t years
• If t years pass, then \(\mathrm{t/2}\) represents the number of doubling periods
• Each doubling multiplies by 2, so after \(\mathrm{t/2}\) periods, we multiply by \(\mathrm{2^{(t/2)}}\)

3. Set up the exponential model

• Starting amount: 275,000
• After t years: \(\mathrm{275,000 \times 2^{(t/2)}}\)
• We want this to equal 1,100,000

4. SIMPLIFY the equation

• Set up: \(\mathrm{1,100,000 = 275,000 \times 2^{(t/2)}}\)
• Divide both sides by 275,000: \(\mathrm{1,100,000 \div 275,000 = 4}\)
• So we have: \(\mathrm{4 = 2^{(t/2)}}\)

5. INFER how to solve the exponential equation

• Recognize that \(\mathrm{4 = 2^2}\)
• So we have: \(\mathrm{2^2 = 2^{(t/2)}}\)
• When bases are equal, exponents must be equal: \(\mathrm{2 = t/2}\)

6. SIMPLIFY to find the final answer

• From \(\mathrm{2 = t/2}\), multiply both sides by 2: \(\mathrm{t = 4}\)
• 4 years after 1985 = 1989

Answer: B. 1989

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often misinterpret "doubles every 2 years" and create the wrong model, such as \(\mathrm{N(t) = 275,000 \times 2^t}\) (doubling every year) instead of \(\mathrm{N(t) = 275,000 \times 2^{(t/2)}}\).

With the wrong model \(\mathrm{2^t}\), they would solve:

\(\mathrm{1,100,000 = 275,000 \times 2^t}\)

getting \(\mathrm{4 = 2^t}\), so \(\mathrm{t = 2}\). This would give them \(\mathrm{1985 + 2 = 1987}\).

This may lead them to select Choice A (1987).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the equation but make arithmetic errors when dividing 1,100,000 by 275,000, or they struggle with the exponential equation solving.

This leads to confusion about the final calculation and may cause them to guess among the remaining choices or select based on partial reasoning.

The Bottom Line:

The key challenge is correctly translating the English phrase "doubles every 2 years" into the mathematical expression \(\mathrm{2^{(t/2)}}\). Students must recognize that t represents total elapsed years, but \(\mathrm{t/2}\) represents the number of doubling periods that occur within those years.