Question

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

Short answer

The residual volume of the dome at a temperature of 35ºC is $160m^3$.

Step-by-step solution

Concept of Volume Expansion in solids.

Volume expansion is the expansion of the dimensions of an object with respect to temperature changes and can be expressed as,

$\mathbf{∆}\mathbf{V}\mathbf{=}{\mathbf{V}}_{\mathbf{0}}\mathbf{\beta }\mathbf{∆}\mathbf{T}$ ...(i)

Here, V is the expansion in volume, V₀ is the unexpanded volume, β is the coefficient of volume expansion and T is the absolute temperature.

 Determination of the diameter value of the rivet following the conditions mentioned.

For Aluminium, the coefficient of volume expansion,

$\beta =7.2\times {10}^{-5}{\left(C°\right)}^{-1}$

The volume of a hemisphere is given by,

$$\begin{gathered} V_0=\frac{2}{3}{\mathrm{\pi R}}^3 \\ =\frac{2}{3}\left(3.14\right){\left(27.5\mathrm{m}\right)}^3 \\ =4.356\times {10}^4{\mathrm{m}}^3 \end{gathered}$$

Substitute all the values in equation (i),

$$\begin{gathered} ∆V=\left(7.2\times {10}^{-5}{\left(C°\right)}^{-1}\right)\left(4.356\times {10}^4{m}^3\right)\left(35C°-\left(-15C°\right)\right) \\ =160m^3 \end{gathered}$$

Thus, the remaining volume of the hemispherical dome is $160m^3$.