A specialized manufacturing process uses the formula \(\mathrm{C(t)} = \frac{8}{5}(\mathrm{t} - 150) + 200\) to calculate the cost per unit...

1. TRANSLATE the problem information

• Given information:
  • Cost function: \(\mathrm{C(t) = \frac{8}{5}(t - 150) + 200}\)
  • Temperature increases by 2.25 degrees Celsius
  • Need to find: increase in cost per unit (not the total cost)

2. INFER the most efficient approach

• This is a linear function in the form \(\mathrm{f(x) = a(x - c) + d}\)
• For linear functions, when input changes by \(\Delta\mathrm{x}\), output changes by \(\mathrm{a}(\Delta\mathrm{x})\)
• The coefficient \(\frac{8}{5}\) tells us the rate of change
• We don't need to calculate specific cost values - just the change

3. SIMPLIFY the calculation

• Coefficient = \(\frac{8}{5}\)
• Temperature change = \(2.25°\mathrm{C}\)
• Cost increase = \(\frac{8}{5} \times 2.25\)
• Calculate: \(8 \times 2.25 = 18.00\)
• Then: \(18.00 \div 5 = 3.60\)

Answer: (A) 3.60

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the linear function property and instead try to calculate specific cost values for two different temperatures.

They might calculate \(\mathrm{C(t)}\) and \(\mathrm{C(t + 2.25)}\) for some specific value of t, leading to unnecessary complexity and potential arithmetic errors. This approach works but is inefficient and error-prone, potentially causing them to select incorrect answers due to calculation mistakes or to abandon the problem entirely.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what "increase" means and calculate the total cost at the new temperature instead of the change in cost.

They might calculate \(\mathrm{C(t + 2.25) = \frac{8}{5}((t + 2.25) - 150) + 200}\) and think this is the answer, potentially leading them to select Choice (C) (236.4) or Choice (D) (436.4) depending on what value they use for t.

The Bottom Line:

This problem tests whether students recognize the efficiency of using linear function properties. The key insight is that you don't need to know the specific temperature - you only need the coefficient and the change in input to find the change in output.