A brick is initially 12.90 cm high and an aluminum can is initially 12.91 cm high. By how much must their temperatures be raised in order for the brick and the can to have exactly the same height?
To find out by how much the temperatures of the brick and aluminum can need to be raised in order for them to have the same height, we can use the concept of thermal expansion.
Thermal expansion is the phenomenon where a substance expands or contracts in response to changes in temperature. Different materials have different coefficients of thermal expansion, which is a measure of how much they expand or contract for each degree change in temperature.
In this case, we need to find the temperature change required for the brick and the can to have the same height. Let's denote this temperature change as ΔT.
The formula to calculate thermal expansion is:
ΔL = α * L0 * ΔT
Where:
ΔL is the change in length or height of the object,
α is the coefficient of linear expansion of the material,
L0 is the initial length or height of the object, and
ΔT is the change in temperature.
Assuming that the coefficient of linear expansion for both the brick and the aluminum can is the same, we can set up the following equation:
α * 12.90 cm * ΔT (for the brick) = α * 12.91 cm * ΔT (for the can)
Here, α can cancel out because we are assuming it is the same for both materials.
Simplifying the equation gives us:
12.90 cm * ΔT (for the brick) = 12.91 cm * ΔT (for the can)
Now, we can solve for ΔT by isolating it on one side of the equation:
12.90 cm * ΔT (for the brick) - 12.91 cm * ΔT (for the can) = 0
ΔT (12.90 cm - 12.91 cm) = 0
ΔT = 0 / (12.90 cm - 12.91 cm)
Since the numerator is zero, the temperature change ΔT can be any value, as long as the denominator is not zero. This means that the temperatures of the brick and the can can never be exactly the same, regardless of how much they are raised or lowered.
Therefore, the answer is that the temperatures of the brick and the can cannot be raised or lowered to have exactly the same height.