A bell tent consists of a conical portion above a cylindrical portion near the ground. For a given volume and a circular base of a given radius, the amount of the canvas used is minimum when the semi-vertical angle of the cone is
A
cos−123
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B
sin−123
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C
cos−113
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D
None of these
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Solution
The correct option is Acos−123 V= Volume of cone + Volume of cylinder =π3r2h+πr2H V=π3r2(h+3H) ⇒H=3Vπr2−h3
Now, surface area, S=πrl+2πrH=πr√h2+r2+2πr×⎛⎜⎝3Vπr2−h3⎞⎟⎠
Now,dSdh=0 ⇒πrh√h2+r2−2πr3=0 ⇒h√h2+r2=23 ⇒5h2=4r2 ⇒rh=√52=tanθ ⇒cosθ=23⇒θ=cos−123