# GATE Papers >> Mechanical >> 2016 >> Question No 163

Question No. 163

For the situation shown in the figure below the expression for $H$ in terms of $r,R$ and $D$ is

##### Answer : (D) $H=(R+r)+\sqrt{2D(R+r)-D^2}$

Solution of Question No 163 of GATE 2016 Mechanical Paper

From diagram

$\begin{array}{l}\mathrm D=\mathrm R+\mathrm{AC}+\mathrm r\\\mathrm{AC}=\mathrm D-\mathrm R-\mathrm r\\\mathrm{AB}=\mathrm R+\mathrm r\end{array}$

$\begin{array}{l}\mathrm{BC}=\sqrt{\mathrm{AB}^2-\mathrm{AC}^2}=\sqrt{\left(\mathrm R+\mathrm r\right)^2-\left(\mathrm D-\left(\mathrm R+\mathrm r\right)\right)^2}\\\mathrm{BC}=\sqrt{\mathrm D\left[2\left(\mathrm R+\mathrm r\right)-\mathrm D\right]}\\\mathrm H=\mathrm R+\mathrm{BC}+\mathrm r\\\mathrm H=\mathrm R+\mathrm r+\sqrt{2\mathrm D\left(\mathrm R+\mathrm r\right)-\mathrm D^2}\end{array}$

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