The radius of a circle increased by 15% from its original length. If the new area is k times the...

1. TRANSLATE the problem information

• Given information:
  • Original radius increases by 15%
  • New area is k times the original area
  • Need to find the value of k

• What this tells us: We need to compare the new area to the original area as a ratio

2. TRANSLATE the percentage increase

• 15% increase means: \(\mathrm{new\ radius = original\ radius + 15\%\ of\ original\ radius}\)
• If original radius = r, then: \(\mathrm{new\ radius = r + 0.15r = 1.15r}\)

3. INFER the approach

• Since we're comparing areas, we need to use the circle area formula for both cases
• The ratio k will be: \(\mathrm{(new\ area) \div (original\ area)}\)

4. SIMPLIFY by calculating both areas

• Original area: \(\mathrm{A_1 = \pi r^2}\)
• New area: \(\mathrm{A_2 = \pi(1.15r)^2}\)

• Let's expand the new area:
  \(\mathrm{A_2 = \pi(1.15)^2r^2}\)
  \(\mathrm{A_2 = \pi(1.3225)r^2}\)
  \(\mathrm{A_2 = 1.3225\pi r^2}\)

5. SIMPLIFY to find k

• Since new area = k × original area:
  \(\mathrm{1.3225\pi r^2 = k \times \pi r^2}\)
• The \(\mathrm{\pi r^2}\) cancels out: \(\mathrm{k = 1.3225}\)

Answer: D (1.3225)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students think that if the radius increases by 15%, the area also increases by 15%, leading them to conclude k = 1.15.

They miss the crucial insight that area depends on radius squared, so when radius is multiplied by 1.15, the area is multiplied by \(\mathrm{(1.15)^2 = 1.3225}\), not just 1.15.

This leads them to select Choice B (1.15).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{(1.15)^2}\) but make arithmetic errors in calculating this value, potentially getting 1.3, 1.45, or other incorrect values.

This may lead them to select Choice C (1.3) or Choice E (1.45).

The Bottom Line:

This problem tests whether students understand that area scaling follows the square of linear scaling. Many students intuitively (but incorrectly) assume that all measurements scale proportionally.