Question

A metal bar with an initial temperature, 𝑇₀, in the interval of 30°C ≤ 𝑇₀ ≤ 35°C

is dropped into a container of boiling water (100°C). The temperature of the 

metal bar, 𝑇 at any time, 𝑡 satisfies the following Newton’s Law of Cooling 

model

𝑑𝑇/𝑑𝑡 = −𝑘(𝑇 − 𝑇_𝑚)

where 𝑇_𝑚 is the ambient temperature and 𝑘 is the constant. After 5 seconds, 

the temperature of the bar, 𝑇₁ is in the interval of 40°C ≤ 𝑇₁ ≤ 50°C.

Find the equation that models the temperature of the metal bar, 𝑇 at 

any time, 𝑡 (choose a value of 𝑇₀ and 𝑇₁ from the given intervals, 

respectively). By using an appropriate analytical method, solve the 

derived model and explain the reason for the selection of the method.

b. Compute the temperature of the metal bar after 100 seconds by using 

the derived model in Part 1(a) with THREE (3) different numerical 

methods with step size, ℎ = 10 seconds. Select the best numerical 

method to compute the temperature of the metal bar and justify your 

answer.