Question

A steel antenna for television broadcasting is \(645.5 \mathrm{~m}\) tall on a summer day, when the outside temperature is \(28.51{ }^{\circ} \mathrm{C} .\) The steel from which the antenna was constructed has a linear expansion coefficient of \(14.93 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1} .\) On a cold winter day, the antenna is \(0.3903 \mathrm{~m}\) shorter than on the summer day. What is the temperature on the winter day?

Short answer

Answer: 92.64°C

Step-by-step solution

Write down the formula for linear expansion and given information

The formula for linear expansion is: \(\Delta L = L_0\alpha\Delta T\) The given information is: - \(L_0 = 645.5 \mathrm{~m}\) - \(\alpha = 14.93 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\) - \(\Delta L = -0.3903 \mathrm{~m}\) - The initial temperature, \(T_0 = 28.51{ }^{\circ} \mathrm{C}\)

Rearrange the formula to solve for \(\Delta T\)

We need to find \(\Delta T\), so rearrange the linear expansion formula as follows: \(\Delta T = \frac{\Delta L}{L_0\alpha}\)

Plug the given values into the equation and solve for \(\Delta T\)

Plugging the given values into the equation, we get: \(\Delta T = \frac{-0.3903 \mathrm{~m}}{(645.5 \mathrm{~m})(14.93 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1})} = -64.13 { }^{\circ} \mathrm{C}\)

Find the temperature on the winter day

Now that we have the change in temperature, we can find the temperature on the winter day by subtracting this value from the initial temperature: \(T_\text{winter} = T_0 - \Delta T = 28.51{ }^{\circ} \mathrm{C} - (-64.13 { }^{\circ} \mathrm{C}) = 92.64{ }^{\circ} \mathrm{C}\) Therefore, the temperature on the winter day is 92.64°C.