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A thin piece of wood of uniform density in the shape of an equilateral triangle with side length $3$ inches weighs $12$ ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length of $5$ inches. Which of the following is closest to the weight, in ounces, of the second piece? (a) 14, (b) 16, (c) 20, (d) 33.3, (e)55.6.

This seems to be a pretty easy problem, but I'm confused because I used the ratios for volume $\frac{3^3}{5^3}$ and got the answer wrong— I was supposed to use areas instead. However, doesn't "weighs 12 ounces" indicate volume?

zoli
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2 Answers2

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The exact volume of such an equilateral triangle of side length $l$ is $$l^2\frac{\sqrt3}2\Delta\times\rho$$ where $\Delta\times \rho$ is the thickness times a factor that transforms the volume to be weight in ounces. We have $$\frac{9\sqrt3}2\Delta\times \rho=12$$ from where $$\Delta\times \rho=\frac{24}{9\sqrt3}.$$

Now, for the other triangle, we have $$\frac{25\sqrt3}2\Delta\times\rho=\frac{25\sqrt3}2\frac{24}{9\sqrt3}=\frac{25\times12}{9}\approx 33.333.$$

zoli
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Since both pieces have the same thickness, the ratio of volumes would be the same as the ratio of areas. Therefore the correct answer is $(12)(5/3)^2 = 100/3.$

Thus the correct ansdwer is (d).$

Mohammad Riazi-Kermani
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