The temperature T in a room varies as a function of the outside temperature T₀ and the number of persons in the room p, according to the relation T = K (θp + T₀), where θ and K are constants. What would be the value of θ, which gives the following data?
T₀
p
T
25
2
32.4
30
5
42.0

Question

The temperature T in a room varies as a function of the outside temperature T₀ and the number of persons in the room p, according to the relation T = K (θp + T₀), where θ and K are constants. What would be the value of θ, which gives the following data?

T₀	p	T
25	2	32.4
30	5	42.0

Prepp Team · Accepted Answer

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.

$ \text{T}_0 $	$ \text{p} $	$ \text{T} $
25	2	32.4
30	5	42.0

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} $:

\[ \text{K} = \frac{32.4}{2\theta + 25} = \frac{32.4}{2(1.0) + 25} = \frac{32.4}{2 + 25} = \frac{32.4}{27} \] \[ \text{K} = 1.2 \]

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.