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From one face of a solid cube of side 14 cm, the largest possible cone is carved out. Find the volume and surface area of the remaining solid. (Use $\pi = \frac{22}{7}, \sqrt{5} = 2.2$)
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Diameter of cone = 14 cm, Radius = 7 cm, Height of cone = 14 cm ($\frac{1}{2} + \frac{1}{2}$ marks). Slant height $l = \sqrt{14^2 + 7^2} = 7\sqrt{5} = 15.4$ cm (1 mark). Volume of remaining solid = Volume of cube - Volume of cone = $(14)^3 - \frac{1}{3} \times \frac{22}{7} \times (7)^2 \times 14 = \frac{6076}{3}$ cm$^3$ (1 + $\frac{1}{2}$ marks). Surface area of remaining solid = Surface area of cube - Area of circle + Curved surface area of cone = $6 \times 14 \times 14 - \frac{22}{7} \times 7 \times 7 + \frac{22}{7} \times 7 \times 15.4 = 1360.8$ cm$^2$ (1 + $\frac{1}{2}$ marks).