Examkin

Question

Difficulty: MediumInterpreting Linear Relationships in Context

A manufacturing plant uses a heating chamber for curing composite materials. The temperature TT, in degrees Celsius (∘C^\circ\text{C}), of the chamber mm minutes after the heating element is turned on is modeled by the equation T=3.5m+22T = 3.5m + 22, where 0≤m≤600 \leq m \leq 60. The chamber must reach a target temperature of 180∘C180^\circ\text{C} for the curing process. Which of the following is the best interpretation of the value 180−223.5\frac{180 - 22}{3.5} in this context?

  1. A
    The rate, in degrees Celsius per minute, at which the temperature of the chamber increases
  2. B
    The temperature, in degrees Celsius, of the heating chamber after 180180 minutes
  3. The number of minutes it takes for the heating chamber to reach the target temperature of 180∘C180^\circ\text{C}Answer
  4. D
    The number of minutes it takes for the temperature of the chamber to increase by 180∘C180^\circ\text{C}

Answer

The number of minutes it takes for the heating chamber to reach the target temperature of 180∘C180^\circ\text{C}
The correct answer represents the time needed to reach the target temperature. To find when the temperature TT reaches 180∘C180^\circ\text{C}, we substitute 180180 for TT in the equation T=3.5m+22T = 3.5m + 22, giving 180=3.5m+22180 = 3.5m + 22. Solving for time mm requires subtracting the initial temperature of 22∘C22^\circ\text{C} from the target temperature of 180∘C180^\circ\text{C}, and then dividing by the heating rate of 3.5∘C3.5^\circ\text{C} per minute. This algebraic isolation results in m=180−223.5m = \frac{180 - 22}{3.5}, representing the duration in minutes to reach that target.

Step-by-Step Solution

1
Identify the meaning of each term in the linear model.
In the equation T=3.5m+22T = 3.5m + 22, TT represents the chamber's temperature, mm represents time in minutes, 3.53.5 represents the rate of temperature increase per minute, and 2222 represents the initial temperature of the chamber.
Understanding the components of the linear equation helps contextualize the mathematical operations performed in the expression.
2
Set up the equation using the target temperature.
The target temperature is 180∘C180^\circ\text{C}. Setting T=180T = 180 gives the linear equation 180=3.5m+22180 = 3.5m + 22.
This establishes the relationship between the target temperature and the time mm required to reach it.
3
Solve for the variable mm representing time.
Subtract 2222 from both sides to get 180−22=3.5m180 - 22 = 3.5m, then divide both sides by 3.53.5 to isolate mm, yielding m=180−223.5m = \frac{180 - 22}{3.5}.
Isolating mm shows that the expression is mathematically equivalent to the time, in minutes, at which the temperature reaches 180∘C180^\circ\text{C}.

Key Concept

Interpreting expressions derived from linear relationships in real-world contexts

Alternative Method

Use dimensional analysis to verify the units of the expression. The numerator (180−22)(180 - 22) represents the difference between two temperatures in degrees Celsius (∘C^\circ\text{C}), which yields a temperature change in ∘C^\circ\text{C}. The denominator, 3.53.5, represents the rate of change in degrees Celsius per minute (∘C/min^\circ\text{C}/\text{min}). Dividing ∘C^\circ\text{C} by ∘C/min^\circ\text{C}/\text{min} results in minutes: ∘C∘C/min=min\frac{^\circ\text{C}}{^\circ\text{C}/\text{min}} = \text{min}. This confirms the expression represents a time duration.
Estimated Time:1m 30s
Rate this question

Topics

Interpreting Linear Relationships in Context

More questions from this topic