document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1138953>Question 1138953</A>: <I><SPAN STYLE=\"word-break: break-all;\"> exponential functions <BR>\n" );
document.write( "The formula for the volume of a cylinder of radius r is V = π r^2h<BR>\n" );
document.write( "a) Solve this equation for r. Write two versions, one in radical form and one in exponential form.<BR>\n" );
document.write( "b) Determine the radius of a cylinder with a volume of 4356π cm^3 and height of 9 cm.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #756729 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=756729\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the formula for the volume of a cylinder is v = pi * r^2 * h\r<BR>\n" );
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document.write( "solve this formula for r.\r<BR>\n" );
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document.write( "divide both sides of the foermula by (pi * h) to get:\r<BR>\n" );
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document.write( "v / (pi * h) = r^2\r<BR>\n" );
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document.write( "flip sides in the equation to get:\r<BR>\n" );
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document.write( "r^2 = v / (pi * h)\r<BR>\n" );
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document.write( "take the square root of both sides of this equaiton to get:\r<BR>\n" );
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document.write( "r = plus or minus sqrt(v / (pi * h)).\r<BR>\n" );
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document.write( "sincer can only be positive, just take the positive square rootto get:\r<BR>\n" );
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document.write( "r = sqrt(v / (pi * h)).\r<BR>\n" );
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document.write( "that's the radical form of the solution.\r<BR>\n" );
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document.write( "since square root is equal to exponent of 1/2, the exponential form of this equation is:\r<BR>\n" );
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document.write( "r = (v / (pi * h)) ^ (1/2).\r<BR>\n" );
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document.write( "to confirm this is correct, do the following.\r<BR>\n" );
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document.write( "let r be equal to any number.\r<BR>\n" );
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document.write( "i used 5.\r<BR>\n" );
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document.write( "let h be equal to any number.\r<BR>\n" );
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document.write( "i used 10.\r<BR>\n" );
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document.write( "v = pi * r^2 * h becomes v = pi * 5^2 * 10 which becomes v = pi * 25 * 10 which becomes v = 250 * pi.\r<BR>\n" );
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document.write( "the formula for r is r = sqrt(v / (pi * h)) in radical form.\r<BR>\n" );
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document.write( "therefore, when v = 250 * pi and h = 10, the formula for r becomes r = sqrt((250 * pi) / (10 * pi).\r<BR>\n" );
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document.write( "simplify this to get r = sqrt(25) because 250/ 10 = 25 and pi/pi is equal to 1.\r<BR>\n" );
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document.write( "solve for r to get r = sqrt(25) = 5.\r<BR>\n" );
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document.write( "you worked your way up with radius equal to 5 and then you were able to work your way back from the volume to get radius = 5, so the formula for the reverse operation was good.\r<BR>\n" );
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document.write( "for part b, you want to find the radius of a cylinder with a volume of 4356 * pi and a height of 9.\r<BR>\n" );
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document.write( "the formula for r we'll use this time is the exponential form of:\r<BR>\n" );
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document.write( "r = (v / (pi * h)) ^ (1/2)\r<BR>\n" );
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document.write( "this formula becomes r = ((4356 * pi) / (pi * 9).\r<BR>\n" );
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document.write( "this can be shown as r = ((4356 / 9) * (pi / pi)).\r<BR>\n" );
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document.write( "simplify to get r = (484) ^ (1/2).\r<BR>\n" );
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document.write( "solve for r to get r = 22.\r<BR>\n" );
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document.write( "working your way back up, you get v = pi * r^2 * h becomes v = pi * 22^2 * 9 which becomes v = pi * 484 * 9 which becomes v = pi * 4356 * pi which confirms the solution is good.\r<BR>\n" );
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document.write( "in exponent form, roots are shown in the denominator of the exponent.\r<BR>\n" );
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document.write( "in other words, x ^ (a/b) is equal to the (b root of x) ^ a or is equal to the b root of (x ^ a).\r<BR>\n" );
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document.write( "consider the cube root of 27 ^ 6.\r<BR>\n" );
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document.write( "in exponential form, this would be 27 ^ (6/3).\r<BR>\n" );
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document.write( "that's equivalent to (the cube root of 27) raided to the 6th power, or equivalent to the cube root of (27 raised to the 6th power).\r<BR>\n" );
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document.write( "if (the cube root of 27) raised to the 6th power, you get 3 raised to the 6th power = 729.\r<BR>\n" );
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document.write( "if the cube root of (27 raised to the 6th power), you get cube root of 387420489 = 729.\r<BR>\n" );
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document.write( "all of this is additional information you may or may not be interested in.\r<BR>\n" );
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document.write( "the solution to your problems is:\r<BR>\n" );
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document.write( "a) Solve this equation for r. Write two versions, one in radical form and one in exponential form.\r<BR>\n" );
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document.write( "r = sqrt(v / (pi * h))\r<BR>\n" );
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document.write( "r = (v / (pi * h)) ^ (1/2)\r<BR>\n" );
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document.write( "b) Determine the radius of a cylinder with a volume of 4356π cm^3 and height of 9 cm.\r<BR>\n" );
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document.write( "the radius of the cylinder is equal to 22 cm.\r<BR>\n" );
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document.write( "here's a reference on radicals.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"https://mathbitsnotebook.com/Algebra1/Exponents/EXFractional.html\" target = \"_blank\">https://mathbitsnotebook.com/Algebra1/Exponents/EXFractional.html</a><BR>\n" );
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