KVPY SX SB Exam 05-Nov-2017 Question Paper

Two bottles A and B have radii ${\mathit{R}}_{\mathit{A}}{\mathrm{and}}_{\mathit{B}}$ and heights ${\mathit{h}}_{\mathit{A}}\mathrm{and}{\mathit{h}}_{\mathit{B}}$ respectively with ${\mathit{R}}_{\mathit{B}}=2{\mathit{R}}_{\mathit{A}}$ and ${\mathit{h}}_{\mathit{B}}=2{\mathit{h}}_{\mathit{A}}$. These are filled with hot water at 60°C. Consider that heat loss for the bottles takes place only from side surfaces. If the time the water to cool down to 50°C is ${\mathit{t}}_{\mathit{A}}\mathrm{and}{\mathit{t}}_{\mathit{B}}$ for the bottles A and B, respectively, then ${\mathit{t}}_{\mathit{A}}\mathrm{and}{\mathit{t}}_{\mathit{B}}$ are best related as