Question:\(\mathrm{T(s) = 0.001(s/60)^3 - 0.122(s/60)^2 + 5.1(s/60) + 22}\)The function T models the temperature, in degrees Celsius, of a certain...

1. TRANSLATE the function notation into context

• Given information:
  • \(\mathrm{T(s)}\) models temperature in degrees Celsius
  • \(\mathrm{s}\) represents seconds after device is powered on
  • We need to interpret \(\mathrm{T(600) ≈ 62}\)
• What this tells us: We have an input of 600 and an output of approximately 62

2. INFER what the input and output mean

• In function notation \(\mathrm{f(x) = y}\):
  • \(\mathrm{x}\) is what goes INTO the function (input)
  • \(\mathrm{y}\) is what comes OUT of the function (output)
• For \(\mathrm{T(600) ≈ 62}\):
  • Input: \(\mathrm{s = 600}\) means 600 seconds after the device is powered on
  • Output: \(\mathrm{T ≈ 62}\) means temperature is approximately 62 degrees Celsius

3. TRANSLATE back to natural language

• Combining the input and output meanings:

  "600 seconds after the device is powered on, the temperature of the component is predicted to be approximately 62 degrees Celsius"

• This matches choice (B) exactly

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students mix up which number represents which quantity, confusing the input and output of the function.

They might think \(\mathrm{T(600) ≈ 62}\) means "the temperature is 600 degrees when the time is 62 seconds" instead of recognizing that 600 is the input (time) and 62 is the output (temperature). This leads them to select Choice A (600 degrees Celsius, 62 seconds).

Second Most Common Error:

Poor context understanding: Students notice that \(\mathrm{s/60}\) appears in the function and calculate \(\mathrm{600/60 = 10}\), then incorrectly use this value as the time instead of the original input value of 600 seconds.

This misconception, combined with potential input/output confusion, leads them to select Choice C (62 degrees Celsius, 10 seconds) or Choice D (10 degrees Celsius, 62 seconds).

The Bottom Line:

Function notation problems require careful attention to what goes in (input) versus what comes out (output), especially when translating between mathematical notation and real-world context. The key is systematically identifying which number represents which quantity based on the function's definition.