Multiple choice

A solid floats with $\dfrac{2}{3}$ of its volume immersed in a liquid and with $\dfrac{3}{4}$ of its volume immersed in another liquid. What fraction of its volume will be immersed if it floats in a homogeneous mixture formed of equal volumes of the liquids?

  1. $\dfrac{6}{7}$
  2. $\dfrac{8}{11}$
  3. $\dfrac{11}{16}$
  4. $\dfrac{12}{17}$
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D Correct answer
Explanation

Let V be volume and d be density of solid. d1, d2 are densities of liquids. V_immersed1 = 2/3 V => d = 2/3 d1. V_immersed2 = 3/4 V => d = 3/4 d2. d1 = 3/2 d, d2 = 4/3 d. Mixture density dm = (d1+d2)/2 = (3/2 + 4/3)/2 * d = (17/6)/2 * d = 17/12 d. Fraction immersed = d/dm = 1 / (17/12) = 12/17.

AI explanation

The fraction of volume immersed is inversely proportional to the density of the liquid, meaning the ratio of the densities of the two liquids is 2 divided by 3 to 3 divided by 4, which simplifies to 8 to 9. When equal volumes of these liquids are mixed, the density of the mixture is the average of their densities, so if the solid has a density of 1 unit and the liquids have densities of 1.5 and 1.333 units, the mixture density is the sum of 0.75 and 0.6665, equaling 1.4165 units. The immersed fraction of the solid's volume in this mixture is its own density divided by the mixture's density, calculated as 1 divided by 1.4165, which equals 12 divided by 17.

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