Multiple choice

The surface area of a closed rectangular prism, which is inscribed in a sphere, is 2976 sq cm, and the sum of the lengths of all its edges is 404 cm. The surface area, in square cm, of the sphere is:

  1. 6056π

  2. 6236π

  3. 7225π

  4. 8181π

  5. a
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C Correct answer
Explanation

For a rectangular prism with dimensions l, w, h: Surface area 2(lw + wh + lh) = 2976. Sum of edges 4(l + w + h) = 404, so l + w + h = 101. The diagonal of the prism is the diameter of the sphere, d^2 = l^2 + w^2 + h^2. We know (l+w+h)^2 = l^2+w^2+h^2 + 2(lw+wh+lh). Thus 101^2 = d^2 + 2976. 10201 = d^2 + 2976, so d^2 = 7225. The surface area of the sphere is 4 * pi * r^2 = pi * d^2 = 7225 * pi.

AI explanation

Let the dimensions of the rectangular prism be l, b, and h. The prism has 12 edges, so the sum of the lengths of all edges is 4(l + b + h) = 404, which means l + b + h = 101. The surface area of the prism is 2(lb + bh + hl) = 2976. The diagonal of the prism is the diameter of the circumscribed sphere, and using the identity (l + b + h)^2 = l^2 + b^2 + h^2 + 2(lb + bh + hl), we have 101^2 = l^2 + b^2 + h^2 + 2976. This gives l^2 + b^2 + h^2 = 10201 - 2976 = 7225, so the square of the diagonal is 7225. The surface area of the sphere is pi times the square of its diameter, which equals 7225pi square cm.

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