Leila is building an aluminum-roofed shed in her backyard to store her garden tools. The flat roof will measure 2.0 x 3.0m in area during the coldest winter months when the temperature is - 10° C, but temperatures in Leila's neighborhood can reach as high as 38° C in the summer. What is the area of the roof that should stick out from the shed in the summer so that the roof just fits the structure during cold winter nights? (aluminum = 24 x

Answer: 0.0138 m^2 = 138 cm^2

Explanation:

The thermal expansion is the term use for the physical phenomena of dilation of the objects when they are exposed to changes in temperature.

The objects dilate when they are heated and contract when they are cooled.

The dilation is proportional to the change in temperatur.

For linear dilation, the proportionality constant is called linear dilation coefficient of the materials, it is named α and is measured in °C ^-1.

ΔL = α * Lo * ΔT, which means that the dilation (or contraction) is proportional to the product of the original length (Lo) and the change of temperature (ΔT).

There is also superficial dilation, for which the dilation is:

ΔA = β * Ao * ΔT, which means that the superficial dilation (or contraction) is proportional to the product of the original area (Ao) and the change of temperature (ΔT).

It is very interesting and important to solve problems that β = 2α, because regularly you will find the values of α for different materials and so, you just to multiply it times 2 to use β.

For this problem:

- Original area, Ao = area of the flat roof at - 10°C = 2.0m * 3.0m = 6.0 m^2.

- α for aluminum = 24 * 10^ - 6 °C^-1.

- ΔT = 38°C - (-10°C) = 48°C

So, ΔA = 6.0m^2 * (2 * 24*10^-6 °C&-1) * 48°C = 0.0138 m^2

And that is the area that should stick out in summer to fit the structure during cold winter nights.

You can pass that number to cm^2 to grasp better the idea of this size:

0.0138 m^2 * (100 cm) ^2 / m^2 = 138 cm^2