Introduction:
In this problem, we are given two meter scales, one made of steel and the other made of aluminum. These scales agree at a temperature of 20 degrees Celsius. We need to calculate the ratio of aluminum centimeters to steel centimeters at three different temperatures: 0 degrees Celsius, 40 degrees Celsius, and 100 degrees Celsius. To solve this problem, we need to use the coefficient of linear expansion for both steel and aluminum.

Explanation:
To calculate the ratio of aluminum centimeters to steel centimeters at different temperatures, we need to consider the change in length of both scales due to temperature variations.

Coefficient of linear expansion:
The coefficient of linear expansion is a measure of how much a material expands or contracts when its temperature changes. It is denoted by the symbol α and has units of per degree Celsius (°C). The coefficient of linear expansion for steel is given as 1.1×10^-5/°C, and for aluminum, it is given as 2.3×10^-5/°C.

Formula for calculating change in length:
The change in length of a material can be calculated using the formula ΔL = α * L0 * ΔT, where ΔL is the change in length, α is the coefficient of linear expansion, L0 is the initial length, and ΔT is the change in temperature.

Calculations:
(a) At 0 degrees Celsius:
The initial length of both the steel and aluminum scales is the same since they agree at 20 degrees Celsius. Therefore, the ratio of aluminum centimeters to steel centimeters at 0 degrees Celsius is 1:1.

(b) At 40 degrees Celsius:
Let's assume the initial length of both scales at 20 degrees Celsius is L0.

For the steel scale:
ΔL_steel = α_steel * L0 * ΔT
ΔL_steel = (1.1 × 10^-5/°C) * L0 * (40°C - 20°C)
ΔL_steel = 0.000022 * L0 * 20
ΔL_steel = 0.00044 * L0

For the aluminum scale:
ΔL_aluminum = α_aluminum * L0 * ΔT
ΔL_aluminum = (2.3 × 10^-5/°C) * L0 * (40°C - 20°C)
ΔL_aluminum = 0.000046 * L0 * 20
ΔL_aluminum = 0.00092 * L0

Therefore, the ratio of aluminum centimeters to steel centimeters at 40 degrees Celsius is:
Ratio = (L0 + ΔL_aluminum) / (L0 + ΔL_steel)
Ratio = (L0 + 0.00092 * L0) / (L0 + 0.00044 * L0)
Ratio = 1.00092 / 1.00044
Ratio ≈ 1.00048

(c) At 100 degrees Celsius:
Using the same calculations as above, we find that the ratio of aluminum centimeters to steel centimeters at 100 degrees Celsius is approximately 1.00116.

Summary:
The ratio of aluminum centimeters to steel centimeters at different