document.write( "Question 65863: 1. Let In(x^2 - 3y) = x - y - 1define a differentiable function y of x. Find an equation of the line tangent to the graph of the equation at the point (2,1)\r
\n" ); document.write( "\n" ); document.write( "2. Water is poured into a conical cup at the rate of 2/3 cubic inches per second. If the cup is 6 inches tall and the top of the cup has a radius of 2 inches, how fast is the water level rising when the water is 4 inches deep? \n" ); document.write( "
Algebra.Com's Answer #46638 by Edwin McCravy(20055)\"\" \"About 
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document.write( "Let ln(x^2 - 3y) = x - y - 1define a differentiable \r\n" );
document.write( "function y of x. Find an equation of the line tangent \r\n" );
document.write( "to the graph of the equation at the point (2,1) \r\n" );
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document.write( "                ln(x² - 3y) = x - y - 1\r\n" );
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document.write( "Differentiate each term implicitly (i.e., without solving \r\n" );
document.write( "for y)\r\n" );
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document.write( "On the left use d[ln(u)]/dx = u'/u\r\n" );
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document.write( "                   2x - 3y'\r\n" );
document.write( "                 ---------- = 1 - y' - 0\r\n" );
document.write( "                   x² - 3y\r\n" );
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document.write( "                   2x - 3y'\r\n" );
document.write( "                 ---------- = 1 - y'\r\n" );
document.write( "                   x² - 3y\r\n" );
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document.write( "Substitute (x.y) = (2,1)\r\n" );
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document.write( "               2(2) - 3y'\r\n" );
document.write( "             -------------- = 1 - y'\r\n" );
document.write( "               (2)² - 3(1)\r\n" );
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document.write( "                    4 - 3y'\r\n" );
document.write( "                  --------- = 1 - y'\r\n" );
document.write( "                     4 - 3\r\n" );
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document.write( "                    4 - 3y'\r\n" );
document.write( "                  --------- = 1 - y'\r\n" );
document.write( "                       1\r\n" );
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document.write( "                    4 - 3y' = 1 - y'\r\n" );
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document.write( "                       -2y' = -3\r\n" );
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document.write( "                         y' = 3/2\r\n" );
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document.write( "So the slope m of the tangent line is m = 3/2\r\n" );
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document.write( "So the tangent line has slope m = 3/2 and passes \r\n" );
document.write( "through the point (2,1) \r\n" );
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document.write( "So we use the point-slope form:\r\n" );
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document.write( "                     y - y1 = m(x - x1)\r\n" );
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document.write( "                      y - 1 = (3/2)(x - 2)\r\n" );
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document.write( "                      y - 1 = (3/2)x - 3\r\n" );
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document.write( "                          y = (3/2)x - 2 \r\n" );
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document.write( "Or if you want it in the standard form of a line:\r\n" );
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document.write( "                   3x - 2y = 4 \r\n" );
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document.write( "----------------------                   \r\n" );
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document.write( "2. Water is poured into a conical cup at the rate of 2/3 \r\n" );
document.write( "cubic inches per second. If the cup is 6 inches tall and \r\n" );
document.write( "the top of the cup has a radius of 2 inches, how fast is \r\n" );
document.write( "the water level rising when the water is 4 inches deep?\r\n" );
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document.write( "Let's assume this is its largest cross-section,\r\n" );
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document.write( " _______________\r\n" );
document.write( " \   2\" |  2\"  /\r\n" );
document.write( "  \     |     / \r\n" );
document.write( "   \    |6\"  / \r\n" );
document.write( "    \   |   /\r\n" );
document.write( "     \  |  / \r\n" );
document.write( "      \ | /    \r\n" );
document.write( "       \|/      \r\n" );
document.write( "             \r\n" );
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document.write( "Now we'll freeze it at some unknown arbitrary water\r\n" );
document.write( "level, x.  Let the radius of the circle which is the\r\n" );
document.write( "circle of the surface of the water level at x be y.\r\n" );
document.write( "     \r\n" );
document.write( " _______________\r\n" );
document.write( " \   2\" |  2\"  /\r\n" );
document.write( "  \     |     / \r\n" );
document.write( "   \    |6\"  / \r\n" );
document.write( "    \___|_y_/_\r\n" );
document.write( "     \  |  / |\r\n" );
document.write( "      \ |x/  |x   \r\n" );
document.write( "       \|/   |   \r\n" );
document.write( "             ¯\r\n" );
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document.write( "The volume of the water at height x and radius of\r\n" );
document.write( "surface circle y is \r\n" );
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document.write( "V = pr²h/3\r\n" );
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document.write( "V = py²x/3\r\n" );
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document.write( "This has 3 unknowns, so we need to reduce the number\r\n" );
document.write( "of unknowns to 2 by finding a way to relate two of them.\r\n" );
document.write( "By similar triangles,\r\n" );
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document.write( "y/x = 2/6\r\n" );
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document.write( "So this gives y = x/3\r\n" );
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document.write( "Substituting that in\r\n" );
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document.write( "V = py²x/3\r\n" );
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document.write( "gives\r\n" );
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document.write( "V = p(x/3)²x/3\r\n" );
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document.write( "V = p(x²/9)x/3\r\n" );
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document.write( "V = px³/27\r\n" );
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document.write( "V = (p/27)x³\r\n" );
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document.write( "Now take the derivative with respect to time t\r\n" );
document.write( "Don't forget the chain rule, i.e., to take the\r\n" );
document.write( "derivative of the \"inside\", when the \"inside\"\r\n" );
document.write( "is not what you are taking the derivative with\r\n" );
document.write( "respect to.\r\n" );
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document.write( "dV/dt = 3(p/27)x²·dx/xt\r\n" );
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document.write( "dV/dt = (p/9)x²·dx/dt\r\n" );
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document.write( "We are given that dV/dt = 2/3 in³/min \r\n" );
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document.write( "So we substitute that and freeze it at x=4,\r\n" );
document.write( "which means to substitute those in:\r\n" );
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document.write( "2/3 = (p/9)4²·dx/dt\r\n" );
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document.write( "2/3 = (16p/9)·dx/dt)\r\n" );
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document.write( "3/(8p) = dx/dt\r\n" );
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document.write( "dx/dt = .1193662073 in/min\r\n" );
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document.write( "Edwin
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