GATE 2026 BT – Question 54

Engineering Mathematics · Differential Equations: First order ODEs, higher order ODEs with constant coefficients, Cauchy and Euler equations, Laplace transforms · 2 marks · Numerical answer

A heat exchanger during operation in a bioprocess has a steady temperature of 90 $^o$C. After completion of its operation, it was shut down and it was observed that the rate of decrease of temperature at any time was directly proportional to the difference $T(t) - 30\,^oC$, where $T(t)$ denotes temperature at time $t$. It was observed that it took 30 min for the temperature to drop to 70 $^o$C. The temperature after 51.5 min will be_________ $^o$C. (rounded off to the nearest integer)

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: 60

Explanation

Newton's law of cooling gives $T - 30 = 60e^{-kt}$. At $t = 30$ min, $40 = 60e^{-30k}$, so $k = \frac{\ln 1.5}{30} = 0.01352$ per min. At $t = 51.5$ min, $T = 30 + 60e^{-0.696} = 30 + 29.9 = 59.9$, which is 60 $^o$C.