Problem $2$: $[18$ points$]$ You are studying the effect of temperature on the dimensions of a blood vessel in the heart of someone with a fever. Assume the blood vessel expands uniformly. It has a linear coefficient of thermal expansion of $\backslash (\backslash$alpha$=1\mathrm{.}5\backslash$times $10^\{-5\}/\{\}^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ It has a volume coefficient of thermal expansion of $\backslash (\backslash$beta$=3\mathrm{.}5\backslash$times $10^\{-5\}/\{\}^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$

a$)[4$ points$]$ A normal human body temperature is $\backslash (37^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ),$ and we can consider $\backslash (39^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ a fever. Provide the values for both of these temperatures in Fahrenheit $(\backslash (\{\}^\{\backslash$circ$\}\backslash$mathrm$\{$F$\}\backslash ))$ and in Kelvin $($K$).$

If you use an online calculator, please provide the formulas that you would use to calculate these temperatures by hand, and cite your calculator $("$Google$"$ is not an appropriate citation for this$).$

b$)[7$ points$]$ You measure the diameter of the vessel to be $3$ mm at a normal body temperature. Calculate the increase in diameter of the blood vessel as the temperature increases from 'normal' to 'fever'.

c$)[7$ points$]$ Instead, if the blood vessel is a cylinder with a diameter of $3$ mm and length of $10$ cm at normal body temperature, calculate the change in the volume of the blood vessel as the temperature increases from 'normal' to 'fever'.