document.write( "Question 552093: Among rectangles with a perimeter of 350 cm, what are the dimensions of the one having maximum area? Lesson is about the application of quadratic equations, and it requires a complete solution. Thank you! :) \n" ); document.write( "

Algebra.Com's Answer #360095 by Theo(13342) $\"\"$ $\"About$

You can put this solution on YOUR website!
a standard form of a quadratic equation has a form of:
\n" ); document.write( "y = ax^2 + bx + c
\n" ); document.write( "a is the coefficient of the x^2 term.
\n" ); document.write( "b is the coefficient of the x term.
\n" ); document.write( "c is the constant term.
\n" ); document.write( "the graph of the quadratic equation looks like an umbrella.
\n" ); document.write( "it's called a parabola.
\n" ); document.write( "if a is positive, the parabola points down and opens up.
\n" ); document.write( "if a is negative, the parabola points up and opens down.
\n" ); document.write( "the first graph below is the parabola pointing down because the coefficient of the x^2 term is positive.
\n" ); document.write( "the second graph below is the parabola pointing up because the coefficient of the x^2 term is negative.
\n" ); document.write( "the equations used are:
\n" ); document.write( "pointing down / opening up:
\n" ); document.write( "y = 2x^2 + 7x - 2
\n" ); document.write( " $\"graph%28400%2C400%2C-5%2C5%2C-10%2C10%2C2x%5E2%2B7x-2%29\"$
\n" ); document.write( "pointing up / opening down:
\n" ); document.write( "y = -2x^2 + 7x - 2
\n" ); document.write( " $\"graph%28400%2C400%2C-5%2C5%2C-10%2C10%2C-2x%5E2%2B7x-2%29\"$
\n" ); document.write( "a parabola has one maximum point or one minimum point.
\n" ); document.write( "if the parabola points down then it has a minimum point.
\n" ); document.write( "if the parabola points up then it has a maximum point.
\n" ); document.write( "how does this apply to your problem?
\n" ); document.write( "see below:
\n" ); document.write( "the perimeter of your rectangle is equal to 350 cm.
\n" ); document.write( "if we let L = length of the rectangle and W = width of the rectangle, then:
\n" ); document.write( "perimeter equals 2L + 2W
\n" ); document.write( "area equals L * W.
\n" ); document.write( "we know that the perimeter 350 cm.
\n" ); document.write( "this means that 2L + 2W = 350
\n" ); document.write( "in this equation, we can solve for W to get:
\n" ); document.write( "W = (350-2L)/2 which can be simplified to:
\n" ); document.write( "W = 275 - L
\n" ); document.write( "if we substitute for W in the equation for the perimeter of the rectangle, then we get:
\n" ); document.write( "2L + 2*(175-L) = 350
\n" ); document.write( "if we let x = L, then this equation becomes:
\n" ); document.write( "2x + 2*(175-x) = 350.
\n" ); document.write( "since we know that the area of the rectangle equals L * W, we can substitute for W in this equation as well to get:
\n" ); document.write( "L * (175-L) = Area of Rectangle.
\n" ); document.write( "if we substitute x for L, then we get:
\n" ); document.write( "x * (175-x) = Area of Rectangle.
\n" ); document.write( "if we let y = area of rectangle, then our equation becomes:
\n" ); document.write( "y = x * (175-x)
\n" ); document.write( "we simplify this equation to get:
\n" ); document.write( "y = 175*x - x^2
\n" ); document.write( "this is a quadratic equation.
\n" ); document.write( "we can graph this equation to get the diagram below:
\n" ); document.write( " $\"graph%28400%2C400%2C-100%2C200%2C-1000%2C10000%2C175%2Ax-x%5E2%29\"$
\n" ); document.write( "you can see that this graph will peak at somewhere around 7500.
\n" ); document.write( "we can solve for the max/min point of the equation by using the formula of:
\n" ); document.write( "x = -b/2a
\n" ); document.write( "our equation is:
\n" ); document.write( "y = 175*x - x^2 which we can re-write as:
\n" ); document.write( "y = -x^2 + 175*x
\n" ); document.write( "this is the standard form of the quadratic equation where:
\n" ); document.write( "a = coefficient of the x^2 term is equal to -1.
\n" ); document.write( "b = coefficient of x term is equal to 175.
\n" ); document.write( "our formula for the max/min point is:
\n" ); document.write( "x = -b/2a which comes out to be:
\n" ); document.write( "x = -175 / (2*(-1)) which comes out to be:
\n" ); document.write( "x = 87.5
\n" ); document.write( "when x = 87.5, the value of y becomes:
\n" ); document.write( "y = -(87.5)^2 + 175*(87.5) which comes out to be:
\n" ); document.write( "y = -7656.25 + 15312.5 which comes out to be:
\n" ); document.write( "y = 7656.25
\n" ); document.write( "since y represents the area of the rectangle, then the maximum area of the rectangle is equal to 7656.25 square cm.
\n" ); document.write( "you can create a table of values to see that this is accurate and that the perimeter will always be 350 cm.
\n" ); document.write( "this table will look as follows:
\n" ); document.write( "

\r\n" );
document.write( "length       width       perimeter        area\r\n" );
document.write( "x            175-x       2x + 2(175-x)    x*(175-x)\r\n" );
document.write( "\r\n" );
document.write( "45           130         350              5850\r\n" );
document.write( "85           90          350              7650\r\n" );
document.write( "87.5         87.5        350              7656.25 ***** (max)\r\n" );
document.write( "90           85          350              7650\r\n" );
document.write( "125          50          350              6250\r\n" );
document.write( "165          10          350              1650\r\n" );
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\n" ); document.write( "this quadratic equation had a maximum value because the coefficient of the x^2 term was negative.\r
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