The function \(\mathrm{F(x) = \frac{9}{5}(x - 273.15) + 32}\) gives the temperature, in degrees Fahrenheit, that corresponds to a temperature...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{F(x) = \frac{9}{5}(x - 273.15) + 32}\) converts kelvins to Fahrenheit
  • Temperature increased by 9.10 kelvins
  • Need to find the increase in Fahrenheit

• What this tells us: We need to compare the Fahrenheit temperature before and after the 9.10 kelvin increase.

2. INFER the approach

• The key insight: "increased by 9.10 kelvins" means if the original temperature was x kelvins, the new temperature is \(\mathrm{(x + 9.10)}\) kelvins
• We need to find \(\mathrm{F(x + 9.10) - F(x)}\) to get the increase in Fahrenheit
• This is a difference calculation, not a direct function evaluation

3. Set up the calculation

• Original Fahrenheit temperature: \(\mathrm{F(x) = \frac{9}{5}(x - 273.15) + 32}\)
• New Fahrenheit temperature: \(\mathrm{F(x + 9.10) = \frac{9}{5}((x + 9.10) - 273.15) + 32}\)
• Increase = \(\mathrm{F(x + 9.10) - F(x)}\)

4. SIMPLIFY the difference expression

• \(\mathrm{F(x + 9.10) - F(x) = [\frac{9}{5}((x + 9.10) - 273.15) + 32] - [\frac{9}{5}(x - 273.15) + 32]}\)
• The +32 terms cancel: \(\mathrm{= \frac{9}{5}((x + 9.10) - 273.15) - \frac{9}{5}(x - 273.15)}\)
• Factor out \(\mathrm{\frac{9}{5}}\): \(\mathrm{= \frac{9}{5}[(x + 9.10 - 273.15) - (x - 273.15)]}\)
• SIMPLIFY inside the brackets: \(\mathrm{= \frac{9}{5}[x + 9.10 - 273.15 - x + 273.15]}\)
• Notice x terms and 273.15 terms cancel: \(\mathrm{= \frac{9}{5}[9.10]}\)

5. Calculate the final result

• \(\mathrm{\frac{9}{5} \times 9.10 = 1.8 \times 9.10 = 16.38}\)

Answer: A. 16.38

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students misunderstand what the problem is asking and think they need to find \(\mathrm{F(9.10)}\) instead of finding the difference \(\mathrm{F(x + 9.10) - F(x)}\).

They substitute x = 9.10 into the original function:

\(\mathrm{F(9.10) = \frac{9}{5}(9.10 - 273.15) + 32}\)

\(\mathrm{= \frac{9}{5}(-264.05) + 32}\)

\(\mathrm{= -475.29 + 32}\)

\(\mathrm{= -443.29}\)

Since this gives a negative result that doesn't match any answer choice, they get confused and might guess. Alternatively, they might take the absolute value and select a choice that seems close.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the problem correctly as \(\mathrm{F(x + 9.10) - F(x)}\) but make algebraic errors during the simplification process.

They might forget to distribute properly or make sign errors when expanding \(\mathrm{(x + 9.10) - 273.15}\), leading to incorrect intermediate calculations that don't simplify to the clean \(\mathrm{\frac{9}{5} \times 9.10}\) result.

This may lead them to select one of the incorrect choices or abandon the algebraic approach and guess.

The Bottom Line:

This problem tests whether students understand that "increase" problems require finding a difference between function values, not just evaluating the function at the given increase amount. The beautiful algebra that cancels most terms is only revealed through correct problem interpretation.