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. Calculate the initial volume of the hemisphere using the formula for the volume of a sphere, \( V = \frac{2}{3} \pi r^3 \), where \( r \) is the radius of the sphere. Since the dome is a hemisphere, use half the diameter as the radius.
2. Determine the change in radius due to thermal expansion using the linear expansion formula, \( \Delta L = L_0 \alpha \Delta T \), where \( L_0 \) is the original length (radius), \( \alpha \) is the coefficient of linear expansion for aluminum, and \( \Delta T \) is the change in temperature from winter to summer.
3. Calculate the new radius in the summer by adding the change in radius, \( \Delta L \), to the original radius.
4. Compute the new volume of the hemisphere in the summer using the updated radius with the formula \( V = \frac{2}{3} \pi r^3 \).
5. Find the difference in volume between the summer and winter by subtracting the initial volume from the new volume calculated for the summer.

Key Concepts

Thermal Expansion

Thermal expansion refers to the tendency of materials to change their dimensions in response to temperature changes. In solids, this typically involves an increase in length, area, or volume as temperature rises. For the aluminum framework of the dome, understanding how it expands with heat is crucial for calculating the change in interior space between winter and summer temperatures.

Volume of a Hemisphere

The volume of a hemisphere can be calculated using the formula V = (2/3)πr³, where r is the radius. This geometric concept is essential for determining the initial volume of the dome at winter temperatures and how it changes with thermal expansion in summer. Knowing the volume allows for a direct comparison of the dome's interior space across different temperatures.

Ideal Gas Law

The Ideal Gas Law (PV = nRT) relates the pressure, volume, and temperature of an ideal gas. While the dome's structure is solid, the air inside can be treated as an ideal gas. Changes in temperature affect the pressure and volume of the air, which can influence the perceived interior space of the dome, especially when considering the expansion of the air as the temperature rises.