A geodesic dome constructed with an aluminum framework is a nearly perfect hemisphere; its diameter measures 55.0 m on a winter day at a temperature of -15°C. How much more interior space does the dome have in the summer, when the temperature is 35°C?

Verified step by step guidance

1. First, understand that the problem involves thermal expansion, which is the increase in volume of a material as its temperature increases. For a solid, the change in volume \( \Delta V \) can be calculated using the formula: \( \Delta V = \beta V_0 \Delta T \), where \( \beta \) is the coefficient of volume expansion, \( V_0 \) is the initial volume, and \( \Delta T \) is the change in temperature.
2. Calculate the initial volume \( V_0 \) of the hemisphere. The formula for the volume of a hemisphere is \( V = \frac{2}{3} \pi r^3 \). First, find the radius \( r \) by dividing the diameter by 2: \( r = \frac{55.0}{2} \) m.
3. Substitute the radius into the volume formula to find \( V_0 \): \( V_0 = \frac{2}{3} \pi (\frac{55.0}{2})^3 \).
4. Determine the change in temperature \( \Delta T \) by subtracting the initial temperature from the final temperature: \( \Delta T = 35°C - (-15°C) = 50°C \).
5. Use the coefficient of volume expansion for aluminum, \( \beta \approx 69 \times 10^{-6} \text{°C}^{-1} \), to calculate the change in volume \( \Delta V \) using the formula: \( \Delta V = \beta V_0 \Delta T \). This will give you the additional interior space in the summer.

Key Concepts

Thermal Expansion

Thermal expansion refers to the tendency of matter to change in volume in response to a change in temperature. For solids like aluminum, this expansion is typically linear, meaning the material expands uniformly in all directions. The coefficient of linear expansion quantifies how much a material expands per degree of temperature change, which is crucial for calculating changes in the dome's dimensions.

Volume of a Hemisphere

The volume of a hemisphere is calculated using the formula V = (2/3)πr³, where r is the radius of the hemisphere. Understanding this formula is essential for determining the interior space of the dome. As the dome's diameter changes due to thermal expansion, the radius changes, affecting the volume calculation.

Temperature Conversion

Temperature conversion is necessary to ensure consistent units when applying formulas involving thermal expansion. In this context, temperatures are given in Celsius, which is suitable for calculating changes in physical properties. Recognizing the temperature difference between winter and summer is key to determining the extent of expansion and its impact on the dome's volume.