document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=250963>Question 250963</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve for r\r<BR>\n" );
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document.write( "S = 1/3(pi)r^2h + 4(pi)rh </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #182774 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=182774\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+%281%2F3%29pi%2Ar%5E2%2Ah+%2B+4pi%2Arh ALIGN=MIDDLE  ALT=\"S+=+%281%2F3%29pi%2Ar%5E2%2Ah+%2B+4pi%2Arh\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(Note: Assuming r is radius and h is height, there is something wrong with this equation. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29pi%2Ar%5E2%2Ah ALIGN=MIDDLE  ALT=\"%281%2F3%29pi%2Ar%5E2%2Ah\"   DISABLED_style= \"max-width:90%; height:auto;\" > would be a number of cubic units and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4pi%2Arh ALIGN=MIDDLE  ALT=\"4pi%2Arh\"   DISABLED_style= \"max-width:90%; height:auto;\" > would be a number of square units. I cannot imagine a context where a sum of cubic units and square units makes any sense. So I will solve this problem using a) an equation that I am guessing is the correct one, and b) the equation you have provided.)<br><BR>\n" );
document.write( "Solving for a variable means \"transform the equation (with proper Algebra) so that that variable is by itself on one side.\" To solve for a variable, it is important to focus on that variable. The rest of the equation is \"clutter\" in the sense that it has no bearing on how we solve for r. Since we're solving for r, we need to focus on the r's in the equation.<br>\r<BR>\n" );
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document.write( "a. (Guessed) Equation: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+%281%2F3%29pi%2Ar%5E2%2Ah+%2B+4pi%2Ar%5E2%2Ah ALIGN=MIDDLE  ALT=\"S+=+%281%2F3%29pi%2Ar%5E2%2Ah+%2B+4pi%2Ar%5E2%2Ah\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Focusing on the r's we should see two <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2 ALIGN=MIDDLE  ALT=\"r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >'s. An equation with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2 ALIGN=MIDDLE  ALT=\"r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > in it is a quadratic equation for r. So we solve for r by solving the quadratic equation.<br><BR>\n" );
document.write( "And how do we solve quadratic equations? The most commonly used way is:<ol><li>Simplify each side of the equation.</li><li>Make one side of the equation zero (by adding and/or subtracting appropriate expressions).</li><li>Next we either use factoring:<ul><li>Factor the non-zero side.</li><li>Use the Zero Product Property and set each factor equal to zero.</li><li>Solve each of the \"factor equals zero\" equations.</li></ul>or use the quadratic formula: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28b%5E2+-+4ac%29%29%2F2a ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28b%5E2+-+4ac%29%29%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" ></li></ol><BR>\n" );
document.write( "You may notice that the phrase \"the most commonly used way [to solve quadratic equations]\" was used. We can use this on the equation. But since there is just <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2 ALIGN=MIDDLE  ALT=\"r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >'s and no r terms, we can use a shorter method:<BR>\n" );
document.write( "Factor out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2 ALIGN=MIDDLE  ALT=\"r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+r%5E2%28%281%2F3%29pi%2Ah+%2B+4pi%2Ah%29 ALIGN=MIDDLE  ALT=\"S+=+r%5E2%28%281%2F3%29pi%2Ah+%2B+4pi%2Ah%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Divide both sides by the other factor:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%2F%28%281%2F3%29pi%2Ah+%2B+4pi%2Ah%29+=+r%5E2 ALIGN=MIDDLE  ALT=\"S%2F%28%281%2F3%29pi%2Ah+%2B+4pi%2Ah%29+=+r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Find the square root of each side:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28S%2F%28%281%2F3%29pi%2Ah+%2B+4pi%2Ah%29%29+=+sqrt%28r%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28S%2F%28%281%2F3%29pi%2Ah+%2B+4pi%2Ah%29%29+=+sqrt%28r%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Normally <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28r%5E2%29+=+abs%28r%29 ALIGN=MIDDLE  ALT=\"sqrt%28r%5E2%29+=+abs%28r%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. But since r represents the length of a radius we can disregard the negative value for r:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28S%2F%28%281%2F3%29pi%2Ah+%2B+4pi%2Ah%29%29+=+r ALIGN=MIDDLE  ALT=\"sqrt%28S%2F%28%281%2F3%29pi%2Ah+%2B+4pi%2Ah%29%29+=+r\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And we have solved for r. The only thing left to do is put the left side in the proper form. We'll multiply the top and bottom of the fraction by 3 to get rid of the fraction within a fraction:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283S%2F%28pi%2Ah+%2B+12pi%2Ah%29%29+=+r ALIGN=MIDDLE  ALT=\"sqrt%283S%2F%28pi%2Ah+%2B+12pi%2Ah%29%29+=+r\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I'll leave it up to you to rationalize the denominator. (Hint: Multiply the top and bottom by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28pi%2Ah+-+12pi%2Ah%29%29 ALIGN=MIDDLE  ALT=\"%28pi%2Ah+-+12pi%2Ah%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >).<br><BR>\n" );
document.write( "b. (Your) equation: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+%281%2F3%29pi%2Ar%5E2%2Ah+%2B+4pi%2Arh ALIGN=MIDDLE  ALT=\"S+=+%281%2F3%29pi%2Ar%5E2%2Ah+%2B+4pi%2Arh\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "With both <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2 ALIGN=MIDDLE  ALT=\"r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and r there are no shortcuts for this one. So we will<BR>\n" );
document.write( "1. Simplify each side. Your equation is already simplified.<BR>\n" );
document.write( "2. Get one side equal to zero (by subtracting S from each side):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+%281%2F3%29pi%2Ar%5E2%2Ah+%2B+4pi%2Arh+-+S ALIGN=MIDDLE  ALT=\"0+=+%281%2F3%29pi%2Ar%5E2%2Ah+%2B+4pi%2Arh+-+S\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using the Commutative Propoerty for Multiplication I will rearrange the terms with r so that the r is at the back:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+%281%2F3%29pi%2Ah%2Ar%5E2+%2B+4pi%2Ahr+-+S ALIGN=MIDDLE  ALT=\"0+=+%281%2F3%29pi%2Ah%2Ar%5E2+%2B+4pi%2Ahr+-+S\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "3. Factoring this is nearly impossible so we will use the Quadratic Formula. With the equation \"a\", the coefficient of the squared term, is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29pi%2Ah ALIGN=MIDDLE  ALT=\"%281%2F3%29pi%2Ah\"   DISABLED_style= \"max-width:90%; height:auto;\" >, \"b\", the coefficient of the r term, is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4pi%2Ah ALIGN=MIDDLE  ALT=\"4pi%2Ah\"   DISABLED_style= \"max-width:90%; height:auto;\" > and \"c\", the term without an r, is S. Using the Quadratic Formula with these expressions for a, b and c we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%28-%284pi%2Ah%29+%2B-+sqrt%28%284pi%2Ah%29%5E2+-+4%28%281%2F3%29pi%2Ah%29%28S%29%29%29%2F%282%28%281%2F3%29pi%2Ah%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(As you can see, the equation which didn't make any sense to begin with also has a very confusing mess of a solution. I will simplify it anyway.)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%28-4pi%2Ah+%2B-+sqrt%2816pi%5E2%2Ah%5E2+-+%284%2F3%29pi%2Ah%2AS%29%29%2F%28%282%2F3%29pi%2Ah%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since we don't leave fractions inside sqaure roots and we don't leave square roots in denominators, I will use a common denominator of 9 inside the square root to subtract:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%28-4pi%2Ah+%2B-+sqrt%28144pi%5E2%2Ah%5E2%2F9+-+%2812%2F9%29pi%2Ah%2AS%29%29%2F%28%282%2F3%29pi%2Ah%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%28-4pi%2Ah+%2B-+sqrt%28%28144pi%5E2%2Ah%5E2+-+12pi%2Ah%2AS%29%2F9%29%29%2F%28%282%2F3%29pi%2Ah%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%28-4pi%2Ah+%2B-+sqrt%28144pi%5E2%2Ah%5E2+-+12pi%2Ah%2AS%29%2Fsqrt%289%29%29%2F%28%282%2F3%29pi%2Ah%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%28-4pi%2Ah+%2B-+sqrt%28144pi%5E2%2Ah%5E2+-+12pi%2Ah%2AS%29%2F3%29%2F%28%282%2F3%29pi%2Ah%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Multiply the numerator and denominator of the \"big\" fraction by 3 (to get rid of the fractions within a fraction):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%28-12pi%2Ah+%2B-+sqrt%28144pi%5E2%2Ah%5E2+-+12pi%2Ah%2AS%29%29%2F%282pi%2Ah%29 ALIGN=MIDDLE  ALT=\"r+=+%28-12pi%2Ah+%2B-+sqrt%28144pi%5E2%2Ah%5E2+-+12pi%2Ah%2AS%29%29%2F%282pi%2Ah%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Again we can reject the negative result leaving:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%28-12pi%2Ah+%2B+sqrt%28144pi%5E2%2Ah%5E2+-+12pi%2Ah%2AS%29%29%2F%282pi%2Ah%29 ALIGN=MIDDLE  ALT=\"r+=+%28-12pi%2Ah+%2B+sqrt%28144pi%5E2%2Ah%5E2+-+12pi%2Ah%2AS%29%29%2F%282pi%2Ah%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "If both your equation and the one I guessed are not the correct equation, I hope some of this will help you figure out a solution. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );