A loaf of bread cools at a constant rate after being removed from the oven. After 5 minutes, its internal...

1. TRANSLATE the problem information

• Given information:
  • Bread cools at a 'constant rate' → linear function \(\mathrm{T(t) = mt + b}\)
  • Two data points: \(\mathrm{(5, 180)}\) and \(\mathrm{(25, 140)}\)
• What we need to find: The specific linear function from the answer choices

2. INFER the solution strategy

• Since we have two points and need a linear function, we must:
  • Find the slope (rate of cooling) first
  • Use one point to find the y-intercept
  • Verify our answer with the second point

3. SIMPLIFY to find the slope

• Using slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• \(\mathrm{m = \frac{140 - 180}{25 - 5}}\)
  \(\mathrm{= \frac{-40}{20}}\)
  \(\mathrm{= -2}\) degrees per minute
• The negative slope makes sense - the bread is cooling down

4. SIMPLIFY to find the y-intercept

• Using point \(\mathrm{(5, 180)}\) in \(\mathrm{T(t) = -2t + b}\):
• \(\mathrm{180 = -2(5) + b}\)
• \(\mathrm{180 = -10 + b}\)
• \(\mathrm{b = 190}\)

5. Write the function and INFER verification is needed

• Our function: \(\mathrm{T(t) = 190 - 2t}\)
• Check with second point: \(\mathrm{T(25) = 190 - 2(25)}\)
  \(\mathrm{= 190 - 50}\)
  \(\mathrm{= 140}\) ✓

Answer: (A) \(\mathrm{T(t) = 190 - 2t}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students see the point \(\mathrm{(5, 180)}\) and incorrectly think 180 is the y-intercept, skipping the proper slope-intercept calculation process.

They might think: 'At 5 minutes it's 180°F, and it's cooling at some rate, so maybe \(\mathrm{T(t) = 180 - 2t}\) works.' Without calculating properly, they don't realize the y-intercept (temperature at \(\mathrm{t = 0}\)) should be 190°F.

This leads them to select Choice (B): \(\mathrm{T(t) = 180 - 2t}\).

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly find the slope as -2 and set up the point-slope form \(\mathrm{T(t) = 190 - 2(t - 5)}\), but fail to distribute and simplify to standard slope-intercept form.

Seeing \(\mathrm{T(t) = 190 - 2(t - 5)}\) matches exactly with choice (D), they select it without recognizing that this form, while mathematically equivalent, doesn't match the standard slope-intercept form the problem expects.

This leads them to select Choice (D): \(\mathrm{T(t) = 190 - 2(t - 5)}\).

The Bottom Line:

This problem tests whether students can systematically work through the linear function process rather than taking shortcuts, and whether they recognize that the y-intercept represents the initial temperature (at \(\mathrm{t = 0}\)), not the temperature at the first given time point.