i) Find the rate of change of temperature at the mid point of the rod that is being heated.

Question

Answer the questions based on the given information. 

Two metal rods, R₁ and R₂, of lengths 16 m and 12 m respectively, are insulated at both the ends. Rod R₁ is being heated from a specific point while rod R₂ is being cooled from a specific point. 

The temperature (T) in Celsius within both rods fluctuates based on the distance (x) measured from either end. The temperature at a particular point along the rod is determined by the equations T = (16 - x)x and T = (x - 12)x for rods R₁ and R₂ respectively, where the distance x is measured in meters from one of the ends. 

i) Find the rate of change of temperature at the mid point of the rod that is being heated. Show your steps.

ii) Find the minimum temperature attained by the rod that is being cooled. Show your work.

Answer 1

i) Identifies that the rod being heated is R₁ and finds the rate of change of temperature at any distance from one end of R₁ as:

\(\frac{dt}{dx}=\frac{d}{dx}(16-x)x=\frac{d}{dx}(16x -x^2)=16-2x\)

Finds the mid-point of the rod as x = 8 m.

Finds the rate of change of temperature at the mid point of R₁ as:

\(\frac{dt}{dx}_{at\,x=8}=16-2(8)=0\)

ii) Identifies that the rod being cooled is R₂ and finds the rate of change of temperature at any distance x m as:

\(\frac{dt}{dx}=\frac{d}{dx} (x-12)x=\frac{d}{dx}(x^2-12x)=2x -12\)

Equates \(\frac{dt}{dx}\) to 0 to get the critical point as x = 6. 

Finds the second derivative of T as:

\(\frac{d^2t}{dx^2}=2\)

And concludes that at x = 6 m, the rod has minimum temperature as

\(\frac{d^2T}{dx^2 ({at\,x=6})}=2>0.\)

Finds the minimum temperature attained by the rod R₂ as 

T(6) = (6 - 12)6 = -36°C.