A steel metre scale and an aluminium metre scale are calibrated to agree exactly at 20^ C . Calculate the ratio aluminium-centimetre/steel-centimetre at 100^ C . The value of for steel is 1.1 10⁻⁵ ^ C⁻¹ while for aluminium it is 2.3 10⁻⁵ ^ C⁻¹ .

Step-by-step solution

Let the true length of 1 cm at 20^ C be L₀ . At 100^ C , the change in temperature is T = 100^ C - 20^ C = 80^ C . The length of 1 cm on the aluminium scale at 100^ C is L_ Al = L₀ (1 + _ Al T) . The length of 1 cm on the steel scale at 100^ C is L_ Steel = L₀ (1 + _ Steel T) . The ratio of aluminium-centimetre to steel-centimetre at 100^ C is: L_ Al L_ Steel = 1 + _ Al T 1 + _ Steel T Using the binomial approximation (1+x)(1+y)⁻¹ 1 + x - y for small x and y : L_ Al L_ Steel 1 + ( _ Al - _ Steel ) T Substituting th