As the drawing shows, two thin strips of metal are bolted together at one end and have the same temperature. One is steel, and the other is aluminum. The steel strip is 0.062% longer than the aluminum strip. By how much should the temperature of the strips be increased, so that the strips have the same length?

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Let's denote the original length of the steel strip at temperature T as L_s and the original length of the aluminum strip as L_a. From the problem, we know that:

L_s = 1.00062 * L_a

When the temperature of both strips is increased by ΔT, their lengths will change according to their linear expansion coefficients. The coefficient of linear expansion for steel (α_s) is approximately 12 x 10^-6 K^-1, and for aluminum (α_a), it's about 24 x 10^-6 K^-1.

So, the change in length for each strip would be:

ΔL_s = α_s * L_s * ΔT
ΔL_a = α_a * L_a * ΔT

The final lengths of the strips after the temperature increase would be:

L_s' = L_s + ΔL_s
L_a' = L_a + ΔL_a

We are looking for a temperature increase, ΔT, that makes L_s' equal to L_a', so:

L_s + ΔL_s = L_a + ΔL_a

Substitute the expressions of ΔL_s and ΔL_a:

L_s + α_s * L_s * ΔT = L_a + α_a * L_a * ΔT

Now substitute the original relationship between L_s and L_a:

1.00062 * L_a + α_s * 1.00062 * L_a * ΔT = L_a + α_a * L_a * ΔT

Now we can factor out L_a and rearrange the terms:

1.00062 + α_s * 1.00062 * ΔT = 1 + α_a * ΔT

Divide both sides by L_a:

1.00062 = 1 + (α_a - α_s * 1.00062) * ΔT

Now solve for ΔT:

ΔT = (1.00062 - 1) / (α_a - α_s * 1.00062)

ΔT = (0.00062) / (24 x 10^-6 - 12 x 10^-6 * 1.00062)

ΔT ≈ (0.00062) / (12 x 10^-6)

ΔT ≈ 51.67 K

So, the temperature of the strips should be increased by about 51.67 K for the strips to have the same length.