A magnesium (Mg) rod and quartz (Qz) rod are laying in your pizzeria’s kitchen a

Question

A magnesium (Mg) rod and quartz (Qz) rod are laying in your pizzeria’s kitchen at 70F. You measure the Mg rod to be 99% as long as the Qz rod. Your pizza oven can reach a top temperature of 800F. Since the lunch rush is over and you’re curious, you place both rods in the hot oven and find that after reaching 800F, the Mg rod is now longer than the Qz rod. At what temperature were the rods the same length? Solve symbolically and then numerically. Note: Mg = 26 × 10^6 Celcius^ 1 and Qz = 0.3 × 10^6 Celcius^1 .

Explanation / Answer

70 degree F=21.1111 degree celcius

800 degree F=426.667 degree celcius

let at 0 degree celcius, length of Qz rod be L1 and Mg rod be L2.

at 21.1111 degree celcius:

99% of Qz rod =length of Mg rod

0.99*L1*(1+0.3*10^(-6)*21.1111)=L2*(1+26*10^(-6)*21.1111)

==>L1=L2*(1+26*10^(-6)*21.1111)/(0.99*(1+0.3*10^(-6)*21.1111))=1.0106*L2

let at a temperature of T degree celcius, both have same length

then L1*(1+0.3*10^(-6)*T)=L2*(1+26*10^(-6)*T)

==>1.0106*L2*(1+0.3*10^(-6)*T)=L2*(1+26*10^(-6)*T)

==>1.0106+1.0106*0.3*10^(-6)*T=1+26*10^(-6)*T

==>0.0106=T*10^(-6)*(26-1.0106*0.3)=25.697*10^(-6)*T

==>T=0.0106/(25.697*10^(-6))=412.5 degree celcius=774.5 degree farenheit

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