Curd is at 80° F, five minutes later it came down at 60°F. After another 5 minutes, its temperature became 50° F. Given that the rate of change of temperature is proportional to (T - S), where S is temperature of the surroundings and T is temperature of the curd at any time t. Then the temperature of the surroundings is :

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The rate of change of temperature follows Newton's Law of Cooling:

$\frac{dT}{dt} = k(T - S)$

where $S$ is the temperature of the surroundings and $T$ is the temperature of the curd at time $t$.

Given information:

• At $t = 0$ minutes: $T = 80°F$
• At $t = 5$ minutes: $T = 60°F$
• At $t = 10$ minutes: $T = 50°F$

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The solution to Newton's Law of Cooling gives:

$T - S = (T_0 - S) \times e^{kt}$

This means the difference from surrounding temperature decreases exponentially.

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Since $e^{10k} = (e^{5k})^2$, the ratio pattern must hold:

$\frac{T_{10} - S}{T_0 - S} = \left[\frac{T_5 - S}{T_0 - S}\right]^2$

Substituting the values:

$\frac{50 - S}{80 - S} = \left[\frac{60 - S}{80 - S}\right]^2$

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$(50 - S)(80 - S) = (60 - S)^2$

$4000 - 50S - 80S + S^2 = 3600 - 120S + S^2$

$4000 - 130S + S^2 = 3600 - 120S + S^2$

$4000 - 130S = 3600 - 120S$

$4000 - 3600 = 130S - 120S$

$400 = 10S$

$S = 40$

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Therefore, the temperature of the surroundings is $40°F$.