{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"math","formal_name":"MATH","introduction":"MATH organises 12,500 high-school competition problems into seven subjects and five difficulty levels, each with a full worked solution. It underlies much of the later mathematics evaluation work; this catalogue imports the full test split: 5,000 problems across all seven subjects.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://github.com/hendrycks/math","indexing_mode":"index","profile":{"resources":[],"task_format":"","scoring":"","metric":"","size":"","answer_access":"","license":"","citation":"","maintainer":"","released":"","why_hard":"","related":[],"history":"","construction":"","limitations":"","saturation":"","scores":[]}},"task_id":"a17d7ffc-f528-5b9a-95c4-f0bf1b602139","task_key":"intermediate~5falgebra--test--a17d7ffc-f528-5b9a-95c4-f0bf1b602139","task_revision_id":"1","upstream_id":"","short_description":"A circular sector has fixed perimeter, but the central angle $\\theta$ and radius…","config":"intermediate_algebra","split":"test","body":"{\"level\":\"Level 4\",\"problem\":\"A circular sector has fixed perimeter, but the central angle $\\\\theta$ and radius $r$ are allowed to vary.\\n\\n[asy]\\nunitsize(2 cm);\\n\\npair A, B, O;\\n\\nA = (1,0);\\nB = dir(110);\\nO = (0,0);\\n\\ndraw(A--O--B);\\ndraw(arc(O,1,0,110));\\n\\nlabel(\\\"$r$\\\", (O + A)/2, S);\\nlabel(\\\"$r$\\\", (O + B)/2, SW);\\nlabel(\\\"$\\\\theta$\\\", (0.1,0.15));\\n[/asy]\\n\\nWhich central angle $\\\\theta$ maximizes the area of the circular sector?  Give the answer in radians.\\n\\nNote: The arc length of the sector is $r \\\\theta,$ and the area of the sector is $\\\\frac{r^2 \\\\theta}{2}.$\",\"type\":\"Intermediate Algebra\"}","display_format":"math","language":"","answer_status":"published","assets":[],"source_url":"https://github.com/hendrycks/math","history":"initial import","indexing_mode":"index","subproblems":[],"grids":[]}