Related Rates Sand-Pile Problem
<Text-field style="Heading 1" layout="Heading 1">Problem 3-17</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Sand pouring from a hopper at a steady rate forms a conical pile whose height is observed to remain twice the radius of the base of the cone. When the height of the pile is observed to be 20 ft, the radius of the base of the pile appears to be increasing at the rate of a foot every two minutes. How fast is the sand pouring from the hopper?
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Solution</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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a1luUmk5R01rWTFSbHh2Ump0R1BVWS9Sa0ZHUkVaSFJrbEdTMFpOUms5R1VVWlRSbFZHWFc4dFJtQnZOajVSSW41R0owWXZSakpHTlVaY2IwWTdSajFHUDBaQlJrUkdSMFpKUmt0R1RVWlBSbEZHVTBaVlJsMXZSbU52Um1WdlJtZHZSbWx2Umx0d1JsMXdSbDl3TDBaaWNGRW1NQzR3WlcxR0p5OUdaWEJHWG5FdFJpdzJOVkVpY2tZblJpOUdNa1kxUmpoR08wWTlSajlHUVVaRVJrZEdTVVpMUmsxR1QwWlJSbE5HVlVaWFJscEdMMFl5UmpWR1hHOUdPMFk5Umo5R1FVWkVSa2RHU1VaTFJrMUdUMFpSUmxOR1ZVWmRidz09Ij5MVWtsYlhKdmQwYzJJeTlKSzIxdlpIVnNaVzVoYldWSE5pSkpMRlI1Y0dWelpYUjBhVzVuUjBrb1gzTjVjMnhwWWtkR0p6WTdMVWtqYldsSFJpUTJOVkVpYUVZbkx5VW5abUZ0YVd4NVIxRXdWR2x0WlhOK1RtVjNmbEp2YldGdVJpY3ZKU1Z6YVhwbFIxRWpNVEpHSnk4bEpXSnZiR1JIVVNabVlXeHpaVVluTHlVbmFYUmhiR2xqUjFFbGRISjFaVVluTHlVcWRXNWtaWEpzYVc1bFIwWTNMeVVxYzNWaWMyTnlhWEIwUjBZM0x5VXNjM1Z3WlhKelkzSnBjSFJIUmpjdkpTdG1iM0psWjNKdmRXNWtSMUVvV3pBc01Dd3dYVVluTHlVclltRmphMmR5YjNWdVpFZFJMbHN5TlRVc01qVTFMREkxTlYxR0p5OGxKMjl3WVhGMVpVZEdOeThsSzJWNFpXTjFkR0ZpYkdWSFJqY3ZKU2x5WldGa2IyNXNlVWRHTnk4bEtXTnZiWEJ2YzJWa1IwWTNMeVVxWTI5dWRtVnlkR1ZrUjBZM0x5VXJhVzF6Wld4bFkzUmxaRWRHTnk4bExIQnNZV05sYUc5c1pHVnlSMFkzTHlVMmMyVnNaV04wYVc5dUxYQnNZV05sYUc5c1pHVnlSMFkzTHlVc2JXRjBhSFpoY21saGJuUkhVU2RwZEdGc2FXTkdKeTFKS0cxbVpXNWpaV1JIUmlRMk5TMUdJelkxTFVZc05qVlJJblJHSjBZdlJqSkdOVVk0Ump0R1BVWS9Sa0ZHUkVaSFJrbEdTMFpOUms5R1VVWlRSbFZHVjBZdlJqSkdOUzlHT1VZM1JqdEdQVVkvUmtGR1JFWkhSa2xHUzBaTlJrOUdVVVpUUmxVdlJsaFJKMjV2Y20xaGJFWW5SaTlHTWtZMVJseHZSanRHUFVZL1JrRkdSRVpIUmtsR1MwWk5SazlHVVVaVFJsVkdYVzh0U1NOdGIwZEdKRFkrVVNJOVJpZEdMMFl5UmpWR1hHOUdPMFk5Umo5R1FVWkVSa2RHU1VaTFJrMUdUMFpSUmxOR1ZVWmRieThsSm1abGJtTmxSMFkzTHlVcWMyVndZWEpoZEc5eVIwWTNMeVVwYzNSeVpYUmphSGxIUmpjdkpTcHplVzF0WlhSeWFXTkhSamN2SlNoc1lYSm5aVzl3UjBZM0x5VXViVzkyWVdKc1pXeHBiV2wwYzBkR055OGxKMkZqWTJWdWRFZEdOeThsSjJ4emNHRmpaVWRSTERBdU1qYzNOemMzT0dWdFJpY3ZKU2R5YzNCaFkyVkhSbU53TFVramJXNUhSaVEyTlZFaU1rWW5SaTlHTWtZMVJseHZSanRHUFVZL1JrRkdSRVpIUmtsR1MwWk5SazlHVVVaVFJsVkdYVzh0Um1Cdk5qNVJJbjVHSjBZdlJqSkdOVVpjYjBZN1JqMUdQMFpCUmtSR1IwWkpSa3RHVFVaUFJsRkdVMFpWUmwxdlJtTnZSbVZ2Um1kdlJtbHZSbHR3Umwxd1JsOXdMMFppY0ZFbU1DNHdaVzFHSnk5R1pYQkdYbkV0Uml3Mk5WRWlja1luUmk5R01rWTFSamhHTzBZOVJqOUdRVVpFUmtkR1NVWkxSazFHVDBaUlJsTkdWVVpYUmxwR0wwWXlSalZHWEc5R08wWTlSajlHUVVaRVJrZEdTVVpMUmsxR1QwWlJSbE5HVlVaZGJ3PT08L0VxdWF0aW9uPjwvUGFyYWdyYXBoPg==Fgt Figure 1 Cross section of sand pile
JSFH Given 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkmbW92ZXJHRiQ2JS1JI21pR0YkNiVRInJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JC1JI21vR0YkNi9RIi5GJy8lJWJvbGRHRjQvRjZRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRkEvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkYvJSlzdHJldGNoeUdGRi8lKnN5bW1ldHJpY0dGRi8lKGxhcmdlb3BHRkYvJS5tb3ZhYmxlbGltaXRzR0ZGLyUnYWNjZW50R0ZGLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVS9GNlEnbm9ybWFsRicvRlJGNC1GOzYtUSI9RidGWEZERkdGSUZLRk1GT0ZRL0ZUUSwwLjI3Nzc3NzhlbUYnL0ZXRmluLUkjbW5HRiQ2JFEiMUYnRlgtRjs2LVEiL0YnRlhGREZHL0ZKRjRGS0ZNRk9GUS9GVFEsMC4xNjY2NjY3ZW1GJy9GV0Zkby1GXG82JFEiMkYnRlhGWA== when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSI9RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUkjbW5HRiQ2JFEjMjBGJ0Y9Rj0=, i.e., when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSI9RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZULUkjbW5HRiQ2JFEjMTBGJ0Y9Rj0= Find LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbW92ZXJHRiQ2JS1JI21pR0YkNiVRIlZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JC1JI21vR0YkNi9RIi5GJy8lJWJvbGRHRjQvRjZRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRkEvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkYvJSlzdHJldGNoeUdGRi8lKnN5bW1ldHJpY0dGRi8lKGxhcmdlb3BHRkYvJS5tb3ZhYmxlbGltaXRzR0ZGLyUnYWNjZW50R0ZGLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGVS9GNlEnbm9ybWFsRicvRlJGNEZY JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Solution Write the volume in terms of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ==. Press the Enter key. Context Menu: Differentiate\342\211\273With Respect To\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Using the evaluation template in the Expression palette, make the following replacements: JSFH 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 (The answer is in cubic feet per minute.) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=