# math - maximize area

by dandre

you have 12 metres of tape what is the largest area you can enclose if you only need to enclose 3 sides

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 Jan 21, 2011 Rating Math - Maximize Area Calculation by: Staff The question: by Dandre (Brampton, Canada, Ontario ) you have 12 metres of tape what is the largest area you can enclose if you only need to enclose 3 sidesThe answer:We can develop an equation for computing the maximum possible area of a rectangle (given your constraints) as follows:A = areaL = lengthW = widthA = L * W3 sides of the rectangle must add up to 12 metresL + W + W = 12Solve for LL + W + W = 12L + 2W = 12L + 2W - 2W = 12 - 2WL + 0 = 12 - 2WL = 12 - 2WSubstitute 12-2W for L in the equation for areaA = L * WA = (12 - 2W) * WA = 12W - 2W^2This is the equation for a parabola which curves downward. Since that is the case, we know the equation will have a maximum value.There are several ways to solve this equation for the width that corresponds to the maximum area.1. Use a table. Use different values of W to calculate the A. Pick the W which produces the highest A.2. Graph the function 12W - 2W^2. Read the value of W which produces the highest A directly off the graph.3. Solve the equation analytically by taking its derivative. Finding the maximum area using a table and a graph can be viewed by clicking on the following link:http://www.solving-math-problems.com/images/math-maximize-area-2011-01-21-for-C2.png3. Solving the equation by taking its derivative is shown below:A = 12W - 2W^2d(A)/dW = d(12W - 2W^2)/dW0 = 12 - 4WSolve for W0 = 12 - 4W0 + 4W = 12 - 4W + 4W0 + 4W = 12 + 04W = 124W/4 = 12/4W*(4/4) = 3W*(1) = 3W = 3The maximum area occurs when the width = 3 metresSolve for LL = 12 - 2WL = 12 - 2*3L = 12 - 6L = 6The final answer is: The maximum area occurs when W = 3 metres and L = 6 metres.The maximum area is 18 square metres.Thanks for writing.Staff www.solving-math-problems.com