Suppose mary would like to design an emergency overheating alarm using one of these materials. in her design, at room temperature (23.0 ∘c)(23.0 ∘c) , a 6.01 cm6.01 cm long cylinder of the material will sit with one end against a wall and the other just barely touching a button. once the heat causes the cylinder to lengthen by 0.0273 cm0.0273 cm , the button will be activated, turning on the alarm. a) at what temperature will the alarm be triggered if the cylinder is made out of polycarbonate? degrees c
b) at what temperature will the alarm be triggered if the cylinder is made out of cast iron? degrees c

A. 92.88 °C

B. 401.535 °C

Explanation:

So, the overheating system is going to be based on the principle of thermal linear expansion. The behavior of this principle is ruled by the equation:

∆L= L_i * ∆T * α  

Where α is a coefficient of linear expansion. For a cylinder made from polycarbonate α = 70,2*10^(-6)  °C^(-1) and for a cylinder made from cast iron α = 12*10^(-6)  °C^(-1). If we isolate the term of the temperature’s difference, we have:

∆L/(L_i * α) = ∆T → T_f = T_i + ∆L/(L_i * α)

Replacing the values, for the case of the Polycarbonate we have:

T_f = T_i + ∆L/(L_i * α) = 23°C+0,0273cm/(6,01cm *70,2 * 10^(-6)°C^(-1) ) = 92,88 °C

Replacing the values, for the case of the Cast Iron we have:

T_f = T_i +∆L/(L_i * α) = 23°C + 0,0273cm/(6,01cm * 12 * 10^(-6) °C^(-1) ) = 401,535 °C

As we see, is way better to use the polycarbonate in this application.

Have a nice day. Let me know if I can help you with anything else. :D