The slant height of a cone with a hemispheric | vedclass

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(D) Let the radius of the hemispherical base be $r$ and the slant height of the cone be $l = 5 \, cm$.The total surface area of the article is the sum

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$7$

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(D) Let the radius of the hemispherical base be $r$ and the slant height of the cone be $l = 5 \, cm$.The total surface area of the article is the sum of the curved surface area of the cone and the curved surface area of the hemisphere.Total Surface Area $= \pi rl + 2\pi r^2 = 103.62 \, cm^2$.Given $\pi = 3.14$,we have $3.14 	imes r 	imes 5 + 2 	imes 3.14 	imes r^2 = 103.62$.$15.7r + 6.28r^2 = 103.62$.Dividing by $3.14$: $5r + 2r^2 = 33$.$2r^2 + 5r - 33 = 0$.Solving the quadratic equation: $2r^2 + 11r - 6r - 33 = 0 \implies r(2r + 11) - 3(2r + 11) = 0$.$(r - 3)(2r + 11) = 0$. Since $r > 0$,$r = 3 \, cm$.The height of the cone $h$ is given by $h = \sqrt{l^2 - r^2} = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4 \, cm$.The total height of the article is the sum of the height of the cone and the radius of the hemisphere: $H = h + r = 4 + 3 = 7 \, cm$.

The slant height of a cone with a hemispherical base is $5 \, cm$. If the total surface area of the article is $103.62 \, cm^2$,find its total height. $(\pi = 3.14)$ (in $cm$)

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