document.write( "Question 163286: 1. Oil is dripping from a pipe at a constant rate and forms a circular pool. The area of the pool is increasing at 15cm^2/s. Find, to 3 significant figures, the rate of increase of the radius of the pool when the area is 50cm^2.\r
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\n" ); document.write( "\n" ); document.write( "3. A particle P moves in a straight line. At time t seconds, the displacement, s metres, of P from a fixed point O of the line is given by s=2tcost+t^2. Find, in m/s to 3 significant figures, the velocity of P when t=3. \n" ); document.write( "
Algebra.Com's Answer #120396 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!

\n" ); document.write( "1. Oil is dripping from a pipe at a constant rate and forms a circular pool. The area of the pool is increasing at 15cm^2/s. Find, to 3 significant figures, the rate of increase of the radius of the pool when the area is 50cm^2.
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document.write( "\"A+=+pi%2Ar%5E2\"\r\n" );
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document.write( "\"%28dA%29%2F%28dt%29=2%2Api%2Ar%28%28dr%29%2F%28dt%29%29\"\r\n" );
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document.write( ">>...The area of the pool is increasing at 15cm^2/s...<<\r\n" );
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document.write( "That says \"%28dA%29%2F%28dt%29=15\".\r\n" );
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document.write( "So we substitute that and we have:\r\n" );
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document.write( "\"15=2%2Api%2Ar%28%28dr%29%2F%28dt%29%29\"\r\n" );
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document.write( "But we also have to substitute \"r\" when \"A=50\"\r\n" );
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document.write( "So we have to calculate \"r\" from \"A=pi%2Ar%5E2\"\r\n" );
document.write( "when \"A=50\" to find out what \"r\" is then.\r\n" );
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document.write( "\"A=pi%2Ar%5E2\"\r\n" );
document.write( "\"50=pi%2Ar%5E2\"\r\n" );
document.write( "\"50%2F%28pi%29=r%5E2\"\r\n" );
document.write( "\"sqrt%2850%2F%28pi%29%29=r\"\r\n" );
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document.write( "So we substitute that in:\r\n" );
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document.write( "\"15=2%2Api%2Ar%28%28dr%29%2F%28dt%29%29\"\r\n" );
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document.write( "\"15=2%2Api%2Asqrt%2850%2F%28pi%29%29%28%28dr%29%2F%28dt%29%29\"\r\n" );
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document.write( "\"15=25.06628275%28%28dr%29%2F%28dt%29%29\"\r\n" );
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document.write( "\"15%2F25.06628275=%28dr%29%2F%28dt%29\"\r\n" );
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document.write( "\".5984134206=%28dr%29%2F%28dt%29\"\r\n" );
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document.write( "Answer: \".598\"\"cm%2Fsec\"\r\n" );
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\n" ); document.write( "2. The region enclosed by the curve with equation \"y%5E2=16x\", the x-axis and the lines x=2 and x=4 is rotated through 360º about the x-axis. Find, in terms of π, the volume of the solid generated.
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document.write( "First we draw the graph of the parabola \"y%5E2=16x\".\r\n" );
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document.write( "Taking square roots, we see this is really two graphs \r\n" );
document.write( "\"y=4sqrt%28x%29\" and \"y=-4sqrt%28x%29\"\r\n" );
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document.write( "Next we'll draw in the vertical lines \"x=2\" and \"x=4\":\r\n" );
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document.write( "Now we'll erase everything that is not involved\r\n" );
document.write( "in the rotation about the x-axis. That leaves only the \r\n" );
document.write( "graph of \"y=4sqrt%28x%29\" between \"x=2\" and \"x=4\"\r\n" );
document.write( "and the x-axis.\r\n" );
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document.write( "We draw a slender rectangle as an element of area\r\n" );
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document.write( "Label the top point of the element (x,y),\r\n" );
document.write( "and the height of it y: \r\n" );
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document.write( "The formula for the volume of a vertically rotated function\r\n" );
document.write( "using the disk method is:\r\n" );
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document.write( "\"V=+pi%2Aint%28%28RADIUS%29%5E2%2C+dx%2C+LEFTMOST_VALUE_OF_x%2C+RIGHTMOST_VALUE_OF_x+%29++\"\r\n" );
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document.write( "The height of that tiny rectangle is y and its width\r\n" );
document.write( "is dx.\r\n" );
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document.write( "It is the height of that rectangle that will rotate \r\n" );
document.write( "about the x-axis, so the radius of rotation is y. The \r\n" );
document.write( "leftmost value of x is 2 and the rightmost value of x \r\n" );
document.write( "is 4.\r\n" );
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document.write( "\"V=+pi%2Aint%28%28y%29%5E2%2C+dx%2C+2%2C+4+%29++\"\r\n" );
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document.write( "Then we replace y by \"4sqrt%28x%29\"\r\n" );
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document.write( "\"V=+pi%2Aint%28%284sqrt%28x%29%29%5E2%2C+dx%2C+2%2C+4+%29++\"\r\n" );
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document.write( "\"V=+pi%2Aint%2816x%2C+dx%2C+2%2C+4+%29++\"\r\n" );
document.write( "\"V=+16pi%2Aint%28x%2C+dx%2C+2%2C+4+%29++\"\r\n" );
document.write( "\"V=+16pi%2A%28x%5E2%2F2%29\"\"matrix%283%2C2%2C%22%7C%22%2C4%2C%22%7C%22%2C%22+%22%2C%22%7C%22%2C2%29\"\r\n" );
document.write( "\"V=+%288pi%29x%5E2\"\"matrix%283%2C2%2C%22%7C%22%2C4%2C%22%7C%22%2C%22+%22%2C%22%7C%22%2C2%29\"\r\n" );
document.write( "\"V=+%288pi%29%284%5E2-2%5E2%29\"\r\n" );
document.write( "\"V=+%288pi%29%2816-4%29\"\r\n" );
document.write( "\"V=+%288pi%29%2812%29\"\r\n" );
document.write( "\"V=+96pi\"\r\n" );
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\n" ); document.write( "3. A particle P moves in a straight line. At time t seconds, the displacement, s metres, of P from a fixed point O of the line is given by \"s=2t%2Acost%2Bt%5E2\". Find, in m/s to 3 significant figures, the velocity of P when t=3.
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document.write( "The velocity of P is the derivative of the displacement s with\r\n" );
document.write( "respect to time t, that is, \"%28ds%29%2F%28dt%29\".\r\n" );
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document.write( "\"s=2t%2Acos%28t%29%2Bt%5E2\"\r\n" );
document.write( "\"%28ds%29%2F%28dt%29=2t%2A%28-sin%28t%29%29%2B+2cos%28t%29%2B2t++\"\r\n" );
document.write( "\"%28ds%29%2F%28dt%29=-2t%2Asin%28t%29%2B2cos%28t%29%2B2t++\"\r\n" );
document.write( "\"%28ds%29%2F%28dt%29=-2%28t%2Asin%28t%29-cos%28t%29-t+%29+\"\r\n" );
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document.write( "When \"t=3\"\r\n" );
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document.write( "\"%28ds%29%2F%28dt%29=-2%283%2Asin%283%29-cos%283%29-3%29\"\r\n" );
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document.write( "When calculating that be sure your calculator \r\n" );
document.write( "is in radian mode, not degree mode.\r\n" );
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document.write( "\"%28ds%29%2F%28dt%29=-0.317\"\"m%2Fs\"\r\n" );
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document.write( "Explanation of the negative sign:\r\n" );
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document.write( "Suppose the line on which P is moving is horizontal.\r\n" );
document.write( "If a positive velocity means that P is moving to the\r\n" );
document.write( "right, then a negative velocity means that P is moving\r\n" );
document.write( "to the left.  So this negative velocity only indicates\r\n" );
document.write( "that at the exact instant when 3 seconds have passed,\r\n" );
document.write( "P is moving left.   \r\n" );
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document.write( "Edwin
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