{"status": "success", "data": {"description_md": "Jesse cuts a circular paper disk of radius $12$ along two radii to form two sectors, the smaller having a central angle of $120$ degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger one?\n\n$\\textbf{(A)}\\ \\frac{1}{8}\\qquad\\textbf{(B)}\\ \\frac{1}{4}\\qquad\\textbf{(C)}\\ \\frac{\\sqrt{10}}{10}\\qquad\\textbf{(D)}\\ \\frac{\\sqrt{5}}{6}\\qquad\\textbf{(E)}\\ \\frac{\\sqrt{5}}{5}$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>Jesse cuts a circular paper disk of radius  <span class=\"katex--inline\">12</span>  along two radii to form two sectors, the smaller having a central angle of  <span class=\"katex--inline\">120</span>  degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger one?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ \\frac{1}{8}\\qquad\\textbf{(B)}\\ \\frac{1}{4}\\qquad\\textbf{(C)}\\ \\frac{\\sqrt{10}}{10}\\qquad\\textbf{(D)}\\ \\frac{\\sqrt{5}}{6}\\qquad\\textbf{(E)}\\ \\frac{\\sqrt{5}}{5}</span> </p>&#10;<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2012 AMC 12B Problem 15", "can_next": true, "can_prev": true, "nxt": "/problem/12_amc12B_p16", "prev": "/problem/12_amc12B_p14"}}