8 Centimeters

Calculus Level 1

The radius, r, of a circle is increasing at a rate of 4 centimeters per minute. Find the rates of change of the area when r = 8 centimeters. Answer is in (cm^2)/min.

32π 64π 16π 128π

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1 solution

Justin Tuazon
Dec 12, 2014

L e t r b e t h e r a d i u s o f t h e c i r c l e L e t t b e t h e t i m e r = 4 t r = 8 , 8 = 4 t t = 2 L e t A b e t h e a r e a o f t h e c i r c l e A = π r 2 S i n c e r = 4 t , A = 16 π t 2 d A d t = 32 π t S i n c e t h e r a i d u s i s 8 w h e n t h e t i m e i s 2 , 32 π ( 2 ) = 64 π T h e r e f o r e , t h e a n s w e r i s 64 π Let\quad r\quad be\quad the\quad radius\quad of\quad the\quad circle\\ Let\quad t\quad be\quad the\quad time\\ \qquad \qquad r=4t\\ r=8,\\ \qquad \qquad 8=4t\\ \qquad \qquad t=2\\ Let\quad A\quad be\quad the\quad area\quad of\quad the\quad circle\\ A=\pi { r }^{ 2 }\\ Since\quad r=4t,\\ \qquad A=16\pi { t }^{ 2 }\\ \frac { dA }{ dt } =32\pi t\\ \\ Since\quad the\quad raidus\quad is\quad 8\quad when\quad the\quad time\quad is\quad 2,\\ \\ 32\pi (2)=64\pi \\ \\ \boxed { Therefore,\quad the\quad answer\quad is\quad 64\pi }

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