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A body cools in a surrounding which is at a constant temperature of $θ_{0}$ . Assume that it obeys Newton's law of cooling. Its temperature $θ$ is plotted against time t. Tangents are drawn to the curve at the points $P (θ = θ_{2})$ and $Q (θ = θ_{1})$ . These tangents meet the time axis at angles of $ϕ_{2}$ and $ϕ_{1}$ , as

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$\frac{\tan ϕ_{2}}{\tan ϕ_{1}} = \frac{θ_{1} - θ_{0}}{θ_{2} - θ_{0}}$

$\frac{\tan ϕ_{2}}{\tan ϕ_{1}} = \frac{θ_{2} - θ_{0}}{θ_{1} - θ_{0}}$

$\frac{\tan ϕ_{1}}{\tan ϕ_{2}} = \frac{θ_{1}}{θ_{2}}$

$\frac{\tan ϕ_{1}}{\tan ϕ_{2}} = \frac{θ_{2}}{θ_{1}}$

answer is B.

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Detailed Solution

$For θ - t plot, rate of cooling = \frac{d θ}{d t} = slope of the curve.$

$At P, \frac{dθ}{dt} = \tan ϕ_{2} = k (θ_{2} - θ_{0}), where k = constant.$

$At Q, \frac{d θ}{d t} = \tan ϕ_{1} = k (θ_{1} - θ_{0})$

$\Rightarrow \frac{\tan ϕ_{2}}{\tan ϕ_{1}} = \frac{θ_{2} - θ_{0}}{θ_{1} - θ_{0}}$

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