Answer to Question #230052 in Calculus for Anuj

Question

Most of you might know that metals expand upon heating and contract when the heating is otherwise. As you are going to be engineers, you also will come to know that dimensions of some objects are really important such that the environmental constraints of places where they are designed and where they’re used are not allowed to vary. Here you have a metal structure that is 10 cm in length and have a certain temperature 70°F will be 𝑦 = 10 + (𝑡 − 70) × 10ିସ Centimeters wide at a nearby temperature 𝑡.  Suppose you need to use this metal in a gravity wave detector, where its width must stay within 0.0005 cm of the ideal 10 cm, how close the temperature 𝑡 should be to 𝑡ⴰ = 70°𝐹 to ensure the tolerance limit is respected?  

Answer 1

Metal structure length = 10 cm

$𝑦 = 10 + (𝑡 − 70) × 10^{-2}\\$

Given that the limit of the metal length is ± 0.0005 cm of 10 cm

So,

$(t-70)10^{-2}=±0.0005 cm\\ (t-70)=±0.05\\ t= 70±0.05 ^0F\\$

The temperature of the metal should be near 70⁰ F and vary only 0.05^oF either more or less. It is not allowed to vary the temperature by more than 0.05^oF to respect the tolerance.

For 70.05⁰ F

$y=10+(70.05-70) \times 10^{-2}\\ y=10.0005 cm$

For 69.95^oF

$y=10+(69.95-70) \times 10^{-2}\\ y=9.9995 cm$