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 tendency of matter to change in volume in response to a change in temperature. For solids like aluminum, this is typically linear expansion.
2. The formula for linear expansion is given by: <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ΔL</mi><mo>=</mo><mi>α</mi><mi>L</mi><mi>ΔT</mi></mrow></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ΔL</mi></math> is the change in length, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> is the coefficient of linear expansion for aluminum, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> is the original length, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ΔT</mi></math> is the change in temperature.
3. Calculate the change in diameter of the dome using the linear expansion formula. The original diameter is 55.0 m, and the temperature change <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ΔT</mi></math> is 50°C (from -15°C to 35°C).
4. Once you have the new diameter, calculate the new radius of the dome by dividing the new diameter by 2. Remember, the dome is a hemisphere, so you will use the formula for the volume of a hemisphere: <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>)</mo><mi>π</mi><msup><mi>r</mi><mn>3</mn></msup></mrow></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> is the radius.
5. Calculate the difference in volume between the winter and summer conditions using the volume formula for a hemisphere. This will give you the additional interior space available 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 geodesic dome. As the dome's diameter changes due to thermal expansion, the radius changes, affecting the volume calculation.

Temperature Conversion and Calculation

Temperature conversion and calculation involve understanding how temperature changes affect physical properties. In this context, converting temperature changes into physical expansion using the material's expansion coefficient is necessary. This concept helps in determining the change in the dome's dimensions from winter to summer, which directly impacts the interior space.