The function \(\mathrm{V(t) = 40,000(0.91)^t}\) models the value, in dollars, of a vehicle t years after it was manufactured. The...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{V(t) = 40,000(0.91)^t}\) models vehicle value
  • Value decreases by \(\mathrm{r\%}\) each year
  • Need to find the value of \(\mathrm{r}\)

2. INFER what the numbers in the function represent

• In exponential functions \(\mathrm{V(t) = P(b)^t}\):
  • \(\mathrm{P}\) = initial value = \(\mathrm{40,000}\)
  • \(\mathrm{b}\) = decay factor = \(\mathrm{0.91}\)
• Key insight: The decay factor tells us what fraction of value remains each year

3. TRANSLATE the decay relationship into mathematics

• If value decreases by \(\mathrm{r\%}\) each year, then:
  • Amount that disappears = \(\mathrm{r\%}\)
  • Amount that remains = \(\mathrm{(100 - r)\%}\)
  • As a decimal: remaining portion = \(\mathrm{\frac{100 - r}{100} = 1 - \frac{r}{100}}\)
• Since decay factor = remaining portion: \(\mathrm{0.91 = 1 - \frac{r}{100}}\)

4. SIMPLIFY to solve for \(\mathrm{r}\)

• Start with: \(\mathrm{0.91 = 1 - \frac{r}{100}}\)
• Isolate \(\mathrm{\frac{r}{100}}\): \(\mathrm{\frac{r}{100} = 1 - 0.91 = 0.09}\)
• Multiply by 100: \(\mathrm{r = 0.09 \times 100 = 9}\)

Answer: C) 9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students confuse the decay factor with the decay rate itself.

They see 0.91 and think "the vehicle loses 0.91% each year" or even "\(\mathrm{r = 0.91}\)." This shows they don't understand that 0.91 represents what REMAINS, not what DISAPPEARS. They might convert 0.91 to a percentage (91%) and think the answer is related to 91.

This may lead them to select Choice B (0.9) thinking \(\mathrm{r = 0.91}\) rounded.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students set up the correct equation \(\mathrm{0.91 = 1 - \frac{r}{100}}\) but make arithmetic errors.

Common mistake: They might incorrectly calculate \(\mathrm{1 - 0.91 = 0.1}\) instead of \(\mathrm{0.09}\), leading to \(\mathrm{r = 10}\). Or they forget to multiply by 100 at the end, giving \(\mathrm{r = 0.09}\).

This may lead them to select Choice A (0.09) by stopping too early in their calculation.

The Bottom Line:

This problem tests whether students truly understand what each part of an exponential decay model represents - specifically that the base tells you what fraction survives each period, not what fraction is lost.