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An isosceles triangle is built from rods: the base is a single rod of length l1 with linear-expansion coefficient alpha1, and the two equal slant sides are rods each of length l2 with coefficient alpha2. The perpendicular distance from the apex to the midpoint of the base (the triangle's height) is to remain unchanged as the temperature varies. What ratio l1/l2 makes this possible?

1. 2*sqrt(alpha2/alpha1)
2. sqrt(alpha2/alpha1)
3. 2*sqrt(alpha1/alpha2)
4. (1/2)*sqrt(alpha2/alpha1)

Correct answer: 2*sqrt(alpha2/alpha1)

Solution

Let h be the apex-to-base-midpoint distance. By Pythagoras h² = l2² - (l1/2)². For h to be temperature independent, d(h²)/dT = 0, i.e. d(l2²)/dT = d((l1/2)²)/dT. Since each length expands as l = l0(1 + alpha*dT), to first order d(l²)/dT = 2*alpha*l². So 2*alpha2*l2² = 2*alpha1*(l1/2)², giving alpha2*l2² = alpha1*l1²/4, hence l1²/l2² = 4*alpha2/alpha1 and l1/l2 = 2*sqrt(alpha2/alpha1).

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