A 53m long steel beam is used to construct the roof of an arena. The beam is installed during the winter when the temperature is -5.0 C. By how much will the beam lengthen in the summer when the temperature reaches 40.0C?

To calculate the change in length of the steel beam, we need to use the coefficient of linear expansion, which measures how much a material expands or contracts when the temperature changes.

The formula to calculate the change in length is:

ΔL = L * α * ΔT

Where:
ΔL is the change in length
L is the original length of the beam
α is the coefficient of linear expansion for steel
ΔT is the change in temperature

First, we need to find the coefficient of linear expansion for steel. The coefficient of linear expansion for steel is approximately 12 x 10^-6 per degree Celsius.

Next, we can substitute the given values into the formula:

L = 53m (original length of the beam)
α = 12 x 10^-6 °C^-1 (coefficient of linear expansion for steel)
ΔT = (40.0 - (-5.0)) = 45.0 °C (change in temperature)

ΔL = 53m * (12 x 10^-6 °C^-1) * 45.0°C

Now, let's solve for ΔL:

ΔL = 53 * 12 x 10^-6 * 45.0
ΔL = 0.02844m

Therefore, the steel beam will lengthen by approximately 0.02844 meters in the summer when the temperature reaches 40.0°C.