Volume of the sphere with radius 1 cm = \(\frac{4}{3} \pi \cdot 1^3 = \frac{4}{3} \pi\) cubic cm Volume of the sphere with radius 6 cm = \(\frac{4}{3} \pi \cdot 6^3 = \frac{4}{3} \pi \cdot 216\) cubic cm Let the outer radius of the hollow sphere be \(r\) cm \(\therefore\) Volume of the hollow sphere = \(\frac{4}{3} \pi [r^3 - (r - 1)^3]\) cubic cm According to the question: \(\frac{4}{3} \pi [r^3 - (r - 1)^3] = \frac{4}{3} \pi + \frac{4}{3} \pi \cdot 216\) i.e., \([r^3 - (r - 1)^3] = 217\) i.e., \(r^3 - (r^3 - 3r^2 + 3r - 1) - 217 = 0\) i.e., \(3r^2 - 3r - 216 = 0\) i.e., \(r^2 - r - 72 = 0\) i.e., \(r^2 - 9r + 8r - 72 = 0\) i.e., \(r(r - 9) + 8(r - 9) = 0\) i.e., \((r - 9)(r + 8) = 0\) \(\therefore\) Either \(r = 9\) or \(r = -8\) But \(r = -8\) is not possible \(\therefore\) The outer radius of the hollow sphere is 9 cm \(\therefore\) Curved surface area of the outer sphere = \(4 \pi \cdot 9^2\) square cm = \(\frac{4 \cdot 22 \cdot 81}{7} = 1018.286\) square cm