GATE2018 CE-1: GA-5

Question

The temperature $T$ in a room varies as a function of the outside temperature $T_0$ and the number of persons in the room $p$, according to the relation $T=K(\theta p +T_0)$, where $\theta$ and $K$ are constants. What would be the value of $\theta$ given the following data?
$$\begin{array}{|c|c|c|} \hline \textbf{$T_0$} & \textbf {p} &\textbf {T}  \\\hline \text{25} & \text{$2$} & \text{$32.4$} \\\hline   \text{$30$}  & \text{$5$} & \text{$42.0$} \\\hline \end{array}$$

• A. 0.8
• B. 1.0
• C. 2.0
• D. 10.0

Answer 1

$T=K(\theta \cdot p +T_0)$

• $32.4=K(2\theta  +25)\qquad \to (1)$
• $42.0=K(5\theta  +30)\qquad \to (2)$

Divide the equation $(1)$ by equation $(2),$ and we get

$\dfrac{32.4}{42.0}=\dfrac{K(2\theta  +25)}{K(5\theta  +30)}$

$\implies\dfrac{324}{420}=\dfrac{2\theta  +25}{5\theta  +30}$

$\implies\dfrac{27}{35}=\dfrac{2\theta  +25}{5\theta  +30}$

$\implies 27(5\theta  +30)=35(2\theta +25)$

$\implies 135\theta +810=70 \theta +875$

$\implies 135\theta -70 \theta =875-810$

$\implies 65\theta = 65$

$\implies \theta = 1$

So, correct answer is $(B).$

Comments

Once we get the ratio as $\frac{27}{35}$, we can use options and mark the answer faster.