Q. 18 - Calculus

Question

Consider a situation in which a substance is heating up in accordance with a model for Newton's Law of Cooling, with a temperature that satisfies $\frac{dT}{dt}=k(350-T)$ for some constant k.

(a) What is the environment's ambient temperature according to this model?

(b) Is the constant k positive or negative, and why, given that the temperature $T\left(t\right)$ is rising and that $0 <T <350$?

(c) Make the case using the differential equation that an object's temperature changes more quickly when it is significantly colder than the surrounding air than when it is more similar to it.

(d) The commonly misunderstood adage "Cold water boils faster" is based on part (c). Why?

Step-by-Step Solution

Answer

1. The ambient temperature is $350°\mathrm{F}.$
   
2. The constant k is positive.
3. The growth rate of temperature will be small.
4. Cold water takes more time to reach the boiling water.

Part(a) Step 1: Given information

The equation $\frac{dT}{dt}=k(350-T)$

Part(a) Step 2: Explanation

The equation $\frac{dT}{dt}=k(350-T)............\left(1\right)$

Remember that the differential equation for the ambient temperature A that models Newton's Law of Cooling and Heating is given by

$\frac{dT}{dt}=k(A-T).............\left(2\right)$

To determine that $A=350$, compare $\left(1\right)$ and $\left(2\right).$ As a result, the ambient temperature is $350°\mathrm{F}.$

Part(b) Step 1: Given information

The temperature is increasing.

Part(b) Step 2: Explanation

The fact that the temperature is rising is obvious. This indicates that the rate of temperature change is increasing. It means that $\frac{dT}{dt}$  is positive or that $\frac{dT}{dt}>0.$ As a result, equation $\left(1\right)$ left side is positive. Since $0<T<350$, the $350-T$ factor is now positive on the right side. Therefore, the constant k must likewise be positive in order for the right hand side to be positive. As a result, the constant k in model $\left(1\right)$ is positive.

Part(c) Step 1: Given information

The factor $A-T$.

Part(c) Step 2: Explanation

The factor $A-T$ determines the increase rate of temperature for a given value of k, and for a predetermined value of $A=350$, the growth rate is solely dependent on the value of $350-T.$ As a result, when $350-T$big values, that is, when $T$has tiny values, the object temperature changes more quickly. In other words, when an object is significantly cooler than the surrounding air, temperature changes occur more quickly. On the other hand, the factor $350-T$ will have small values when $T$ is close to the ambient temperature, indicating that the pace of temperature increase will be modest.

Part(d) Step 1: Given information

Cold water boils faster.

Part(d) Step 2: Explanation

According to the justification in subparagraph (c), the rate of temperature increase is greater when the water is cold, but it will take longer for the temperature to reach the boiling point when the water is cold than it will when it is warm. Therefore, the faster rate of temperature increase for cold water is misconstrued in terms of the amount of time needed to reach the boiling point.