# Find the rate of change of temperature with time.

The temperature distribution at a certain input of time in concrete slab during curing is given by $$T=3x^{2}+3x+16$$ where $$x$$ is in $$cm$$ and $$T$$ is in $$K$$.Find the rate of change of temperature with time. $$(\alpha=0.003\;cm^{2}/s)$$

Asked on 25th October 2019 in

The rate of change of temperature with time is given as $\frac{dT}{dt}$

Here $$T=3x^{2}+3x+16$$

For a 1-D unsteady heat flow without any internal heat generation

$\frac{d^{2}T}{dx^{2}}=\frac{1}{\alpha}\frac{dT}{dt}$

$$\frac{dT}{dx}=\frac{d}{dx}\left ( 3x^{2}+3x+16 \right ) \\=6x+3 \\\frac{d^{2}T}{dx^{2}}=\frac{d}{dx}\left ( 6x+3 \right ) \\=6$$

$$\Rightarrow 6=\frac{1}{\alpha}\frac{dT}{dt} \\\Rightarrow 6=\frac{1}{0.003}\frac{dT}{dt} \\\Rightarrow \frac{dT}{dt}=6\times 0.003 = 0.018\;K/s$$