What is the maximum volume expansion coefficient of ?

[castrodisastro]

Homework Statement

You are building a device for monitoring ultracold environments. Because the device will be used in environments where its temperature will change by 211°C in 2.99s, it must have the ability to withstand thermal shock (rapid temperature changes). The volume of the device is 3.00⋅10⁻⁵m³, and if the volume changes by 1.00⋅10⁻⁷m³ in a time interval of 7.15s, the device will crack and be rendered useless. What is the maximum volume expansion coefficient that the material you use to build the device can have?

ΔT = 211 °C
V₀ = 3.00⋅10^-5 m³
ΔV = 1.00⋅10^-7 m³
β = ?

Homework Equations

ΔV = β(ΔT)V₀

The Attempt at a Solution

It seems like I am given everything to calculate the volume expansion coefficient, β.

I am not sure how the time limit of 2.99 s comes into play here if it takes us longer than 2.99 s for the temperature to change so the risk of thermal shock is avoided and seems like extra information and not something I need to take in account. I realize I may be wrong and want to understand why.

I rearranged to solve β

β = (ΔV)/(ΔT)(V₀)

β = (1.00⋅10^-7 m³)/(211 °C)(3.00⋅10^-5 m³)

β = 1.5798⋅10^-5

I submitted this problem to my online homework and I was incorrect.

Any help would be appreciated.

 

[mfb]

The given time values are confusing, I agree.

Maybe the cracking limit has to be seen as volume per time, so in 2.99 seconds the maximal volume change is just 2.99/7.15 of the given value.
On the other hand, cooling won't be uniform in general, so this is a bit unrealistic.

 

[castrodisastro]

I agree. For some problems to be more realistic would require differential equations, but our book tries to simplify these problems by assuming constant values for certain things like volume that would require them. Our professor didn't say anything specifically about this problem however.