Multiple choice

The sum of the edge lengths of two cubes is 10 units, and the sum of the volumes of both the cubes is 280 cubic units. Find the sum of the surface areas of both the cubes.

  1. 90 square units

  2. 100 square units

  3. 216 square units

  4. 312 square units

  5. 512 square units
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D Correct answer
Explanation

Let edges be x and y. x+y = 10, x^3 + y^3 = 280. x^3 + y^3 = (x+y)(x^2 - xy + y^2) = 10((x+y)^2 - 3xy) = 280. 100 - 3xy = 28. 3xy = 72, xy = 24. x^2 + y^2 = (x+y)^2 - 2xy = 100 - 48 = 52. Sum of surface areas = 6x^2 + 6y^2 = 6(52) = 312.

AI explanation

Let the edge lengths of the two cubes be x and y. We are given x + y = 10 and x^3 + y^3 = 280. Using the identity x^3 + y^3 = (x + y)^3 - 3xy(x + y), we substitute the known values to get 280 = 1000 - 30xy, which gives xy = 24. The total surface area of both cubes is 6x^2 + 6y^2 = 6(x^2 + y^2). Using the identity x^2 + y^2 = (x + y)^2 - 2xy, we find x^2 + y^2 = 100 - 48 = 52. The sum of the surface areas is 6 multiplied by 52, which equals 312 square units.

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