Mathematics

Mathematics

Problem

A pot of soup is removed from the stove at 95∘C95^{\circ}\text{C} and placed in a 20∘C20^{\circ}\text{C} kitchen. If its temperature satisfies dTdt=k (T−20)\displaystyle \frac{dT}{dt}=k\,(T-20) with k=−0.08 min−1k=-0.08\,\text{min}^{-1}, which is the best estimate of the soup’s temperature after 1010 minutes? a) 43∘C43^{\circ}\text{C} b) 70∘C70^{\circ}\text{C} c) 54∘C54^{\circ}\text{C} d) 50∘C50^{\circ}\text{C}
Intuitive explanation

This is a separable differential equation modeling Newton's Law of Cooling—the rate of temperature change is proportional to how far the soup is from room temperature. The key insight is to separate variables and integrate: rearrange to dTT−20=k dt\frac{dT}{T-20} = k\,dt, then use your initial condition T(0)=95T(0) = 95 to find the constant of integration. Once you have the explicit formula for T(t)T(t), plug in t=10t=10 to get your answer.

Intuitive explanation

This is a separable differential equation modeling Newton's Law of Cooling—the rate of temperature change is proportional to how far the soup is from room temperature. The key insight is to separate variables and integrate: rearrange to dTT−20=k dt\frac{dT}{T-20} = k\,dt, then use your initial condition T(0)=95T(0) = 95 to find the constant of integration. Once you have the explicit formula for T(t)T(t), plug in t=10t=10 to get your answer.

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