Temperature Relation Analysis

The problem presents a relation describing the temperature T in a room. This room temperature T depends on the outside temperature T₀ and the number of persons in the room p. The given mathematical relation is \( \text{T} = \text{K} (\theta\text{p} + \text{T}_0) \), where \( \theta \) and \( \text{K} \) are constants. Our goal is to determine the value of \( \theta \) using the provided experimental data.

Data Interpretation for Temperature Calculation

We are given two sets of data points, each showing a specific room temperature T corresponding to a particular outside temperature T₀ and number of persons p. This data is crucial for forming equations to solve for the unknown constants.

Setting Up the Temperature Equations

We can substitute each set of data into the given temperature relation \( \text{T} = \text{K} (\theta\text{p} + \text{T}_0) \) to form two linear equations. These equations will involve the two unknown constants, \( \text{K} \) and \( \theta \).

• Equation 1 (Using Data Set 1: \( \text{T}_0 = 25, \text{p} = 2, \text{T} = 32.4 \)): \[ 32.4 = \text{K} (\theta(2) + 25) \] \[ 32.4 = \text{K} (2\theta + 25) \quad \ldots(1) \]
• Equation 2 (Using Data Set 2: \( \text{T}_0 = 30, \text{p} = 5, \text{T} = 42.0 \)): \[ 42.0 = \text{K} (\theta(5) + 30) \] \[ 42.0 = \text{K} (5\theta + 30) \quad \ldots(2) \]

Solving for Theta ($\theta$) in the Temperature Equation

Now we have a system of two equations with two unknowns, \( \text{K} \) and \( \theta \). We can solve for \( \theta \) by first expressing \( \text{K} \) in terms of \( \theta \) from one equation and then substituting it into the other.

• From Equation (1), we can express \( \text{K} \): \[ \text{K} = \frac{32.4}{2\theta + 25} \]
• Substitute this expression for \( \text{K} \) into Equation (2): \[ 42.0 = \left(\frac{32.4}{2\theta + 25}\right) (5\theta + 30) \]
• Multiply both sides by \( (2\theta + 25) \) to eliminate the denominator: \[ 42.0 (2\theta + 25) = 32.4 (5\theta + 30) \]
• Distribute the constants on both sides: \[ 84\theta + 1050 = 162\theta + 972 \]
• Gather the \( \theta \) terms on one side and the constant terms on the other: \[ 1050 - 972 = 162\theta - 84\theta \] \[ 78 = 78\theta \]
• Finally, solve for \( \theta \): \[ \theta = \frac{78}{78} \] \[ \theta = 1.0 \]

Calculating the Constant K (Optional Verification)

Although the question only asks for \( \theta \), we can find the value of \( \text{K} \) to ensure consistency and for a complete understanding of the relationship. Substitute the calculated value of \( \theta = 1.0 \) back into the expression for \( \text{K} \):

Final Temperature Verification

To confirm our values, we can plug \( \theta = 1.0 \) and \( \text{K} = 1.2 \) back into the original relation for both data sets.

• For Data Set 1: \( \text{T}_0 = 25, \text{p} = 2 \) \[ \text{T} = 1.2 (1.0(2) + 25) = 1.2 (2 + 25) = 1.2 (27) = 32.4 \] This matches the given temperature.
• For Data Set 2: \( \text{T}_0 = 30, \text{p} = 5 \) \[ \text{T} = 1.2 (1.0(5) + 30) = 1.2 (5 + 30) = 1.2 (35) = 42.0 \] This also matches the given temperature.

Both verifications confirm that the calculated value of \( \theta = 1.0 \) is correct.