Question

Finding the Original Temperature of a Beam An aluminum beam was brought from the outside cold into a machine shop where the temperature was held at \(65^{\circ} \mathrm{F}\) . After 10 min, the beam warmed to \(35^{\circ} \mathrm{F}\) and after another 10 \(\mathrm{min}\) its temperature was \(50^{\circ} \mathrm{F} .\) Use Newton's Law of Cooling to estimate the beam's initial temperature.

Short answer

The initial temperature of the aluminum beam, assuming Newton's law of cooling, is equal to the value found for \(T_0\)

Step-by-step solution

Set up the equation for the first temperature increase

Using Newton's law of cooling formula, we can write the equation for the first 10 minute interval: \(35^{\circ} F = 65^{\circ} F + (T_0 - 65^{\circ} F) e^{-10k}\). Let's simplify this equation to isolate \(k\) and \(T_0\)

Solve for $k$ from the first equation

Rearrange the equation, isolate \(k\), and use logarithms to solve for \(k\). The result will be an expression for \(k\) in terms of \(T_0\)

Set up the equation for the second temperature increase

Applying Newton's law of cooling for second 10 minute interval, we can write the equation as: \(50^{\circ} F = 65^{\circ} F + (T_0 - 65^{\circ} F) e^{-20k}\). Then, substitute the value of \(k\) obtained from Step 2 into this equation.

Solve for $T_0$

After substituting the value of \(k\), we have an equation in terms of \(T_0\) only. It'll be a exponential equation that we can now solve for \(T_0\)

Calculate the initial temperature

Use the solution for \(T_0\) from Step 4 in the formula derived in Step 2 to get the actual value for \(T_0\)