document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=114393>Question 114393</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose a rectangular movie screen has an area of 10416 square feet. Find the dimensions of the screen if it is 19 feet longer than it is wide.\r<BR>\n" );
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document.write( "THANKS! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #83290 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=83290\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First let L represent the length of the movie screen and let W represent the width of the screen.<BR>\n" );
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document.write( "You are told that the length is 19 feet longer than the width. So you can say in equation form<BR>\n" );
document.write( "that:<BR>\n" );
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document.write( "L = W + 19<BR>\n" );
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document.write( "You are also told that the area of the screen is 10416 sq ft. But the area of a rectangle<BR>\n" );
document.write( "such as this screen is is found by multiplying its length L by its width W. So we can write<BR>\n" );
document.write( "in equation form that:<BR>\n" );
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document.write( "Area = L * W<BR>\n" );
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document.write( "Substitute 10416 for A and W + 19 for L and you have:<BR>\n" );
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document.write( "10416 = (W + 19)*W<BR>\n" );
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document.write( "Multiply out the right side by multiplying W times each of the terms in the parentheses<BR>\n" );
document.write( "and the equation becomes:<BR>\n" );
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document.write( "10416 = W^2 + 19W<BR>\n" );
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document.write( "Get this into standard quadratic form by subtracting 10416 from both sides to make the<BR>\n" );
document.write( "equation:<BR>\n" );
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document.write( "0 = W^2 + 19W - 10416<BR>\n" );
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document.write( "Just to get it into a more conventional form, transpose the equation (switch sides) to<BR>\n" );
document.write( "make it:<BR>\n" );
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document.write( "W^2 + 19W - 10416 = 0<BR>\n" );
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document.write( "This is in the standard quadratic form of:<BR>\n" );
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document.write( "aW^2 + bW + c = 0<BR>\n" );
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document.write( "and by comparing this standard form with the conventional for for this problem, we can see<BR>\n" );
document.write( "that a = 1, b = 19, and c = -10416<BR>\n" );
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document.write( "For the standard quadratic form, according to the quadratic formula the answers for the <BR>\n" );
document.write( "unknown are:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"W+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Substitute 1 for a, 19 for b, and -10416 for c and you have:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W+=+%28-19+%2B-+sqrt%28+19%5E2-4%2A1%2A-10416+%29%29%2F%282%2A1%29+ ALIGN=MIDDLE  ALT=\"W+=+%28-19+%2B-+sqrt%28+19%5E2-4%2A1%2A-10416+%29%29%2F%282%2A1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Solve for the term under the radical. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19%5E2+=+361 ALIGN=MIDDLE  ALT=\"19%5E2+=+361\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2A1%2A-10416+=+41664 ALIGN=MIDDLE  ALT=\"-4%2A1%2A-10416+=+41664\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Add<BR>\n" );
document.write( "these two terms to get that the term under the radical is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=361+%2B+41664+=+42025 ALIGN=MIDDLE  ALT=\"361+%2B+41664+=+42025\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "This reduces the answer to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W+=+%28-19+%2B-+sqrt%28+42025+%29%29%2F%282%2A1%29+ ALIGN=MIDDLE  ALT=\"W+=+%28-19+%2B-+sqrt%28+42025+%29%29%2F%282%2A1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Taking the square root of 42025 on a calculator we get 205. Substituting this result further<BR>\n" );
document.write( "reduces the equation for W to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W+=+%28-19+%2B-+205%29%2F%282%2A1%29+ ALIGN=MIDDLE  ALT=\"W+=+%28-19+%2B-+205%29%2F%282%2A1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Multiply out the denominator ... 2*1 = 2 and this simplifies the equation to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W+=+%28-19+%2B-+205%29%2F2+ ALIGN=MIDDLE  ALT=\"W+=+%28-19+%2B-+205%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The numerator must be positive so that the value of W is positive. [It would make no sense<BR>\n" );
document.write( "to have a negative width.] Therefore, we can disregard the negative sign preceding the<BR>\n" );
document.write( "205 and the equation becomes just:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W+=+%28-19+%2B+205%29%2F2+ ALIGN=MIDDLE  ALT=\"W+=+%28-19+%2B+205%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The two terms in the numerator combine to 186 and the equation is simplified to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W+=+186%2F2+=+93 ALIGN=MIDDLE  ALT=\"W+=+186%2F2+=+93\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This tells you that the small dimension of the screen is 93 feet. The larger dimension<BR>\n" );
document.write( "(the length) is 19 feet more or 93 + 19 = 112 ft.<BR>\n" );
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document.write( "So the screen is 93 by 112 feet.<BR>\n" );
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document.write( "Check ... the area of the screen is 93 * 112 = 10416 square feet, just as the problem<BR>\n" );
document.write( "says it should be. Our answer checks.<BR>\n" );
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document.write( "Hope this helps you to understand the problem and how to use the quadratic formula in<BR>\n" );
document.write( "solving it.<BR>\n" );
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