A metallic sphere cools from $50^{0}$ C to $40^{0}$ C in 300s. If atmospheric temperature around is $20^{0}$ C, then the sphere’s temperature after the next 5 minutes will be close to :

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$35^{0} C$

$33^{0} C$

$28^{0} C$

$31^{0} C$

answer is B.

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Detailed Solution

$N e w t o n' s L a w o f C o o l i n g,$

$\begin{array}{l} \begin{array}{l} \frac{θ_{1} - θ_{2}}{t_{1} - t_{2}} = k (\frac{θ_{1} + θ_{2}}{2} - θ_{s u r}) \\ \Rightarrow \frac{50 - 40}{300} = k (\frac{50 + 40}{2} - 20) \\ \Rightarrow \frac{10}{300} = k (45 - 20) - - - (1) \end{array} \\ \begin{array}{l} N o w, \frac{40 - θ_{f}}{300} = k (\frac{40 + θ_{f}}{2} - 20) - - - - - (2) \\ D i v i d i n g (1) b y (2), \frac{10}{40 - θ_{f}} = \frac{25}{(\frac{40 + θ_{f}}{2} - 20)} \end{array} \\ \begin{array}{l} \Rightarrow \frac{10}{40 - θ_{f}} = \frac{25}{θ_{f}} \times 2 \\ \Rightarrow θ_{f} = \frac{100}{3} = 33^{0} C \end{array} \end{array}$