Cone

Geometry Level 2

The figure above was the result after a right circular cone was cut by a plane parallel to the base. The upper base has an area of 9 4 π \dfrac{9}{4}\pi while the lower base has an area of 49 4 π \dfrac{49}{4}\pi . If the perpendicular distance between the bases is 5 5 , what is the lateral area? If your answer can be expressed as a π b a\pi \sqrt{b} , where a a and b b are positive coprime integers and b b is square free, give your answer as a + b a+b .

25 36 29 34

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1 solution

From the formula of the area of a circle, A = π r 2 A=\pi r^2 , the diameter of the upper base is 3 3 and the lower base is 7 7 .

l a t e r a l a r e a = 1 2 ( s u m o f t h e c i r c u m f e r e n c e s o f t h e b a s e s ) ( s l a n t h e i g h t ) lateral~area=\dfrac{1}{2}(sum~of~the~circumferences~of~the~bases)(slant~height)

l a t e r a l a r e a = 1 2 ( 3 π + 10 π ) ( 5 2 + 2 2 ) = 10 2 π ( 29 ) = 5 π 29 ) lateral~area=\dfrac{1}{2}(3\pi + 10\pi)(\sqrt{5^2+2^2})=\dfrac{10}{2}\pi (\sqrt{29})=5\pi\sqrt{29})

Finally, the answer desired is a + b = 5 + 29 = a+b=5+29= 34 \color{#D61F06}\boxed{34}

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