logo for solving-math-problems.com
leftimage for solving-math-problems.com

math - maximize area

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 sides

Comments for math - maximize area

Average Rating starstarstarstarstar

Click here to add your own comments

Jan 21, 2011
Rating
starstarstarstarstar
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 sides




The answer:


We can develop an equation for computing the maximum possible area of a rectangle (given your constraints) as follows:

A = area
L = length
W = width

A = L * W

3 sides of the rectangle must add up to 12 metres

L + W + W = 12

Solve for L

L + W + W = 12

L + 2W = 12

L + 2W - 2W = 12 - 2W

L + 0 = 12 - 2W

L = 12 - 2W

Substitute 12-2W for L in the equation for area


A = L * W

A = (12 - 2W) * W

A = 12W - 2W^2

This 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.png


3. Solving the equation by taking its derivative is shown below:


A = 12W - 2W^2

d(A)/dW = d(12W - 2W^2)/dW

0 = 12 - 4W


Solve for W

0 = 12 - 4W

0 + 4W = 12 - 4W + 4W

0 + 4W = 12 + 0

4W = 12

4W/4 = 12/4

W*(4/4) = 3

W*(1) = 3

W = 3

The maximum area occurs when the width = 3 metres

Solve for L

L = 12 - 2W

L = 12 - 2*3

L = 12 - 6

L = 6

The 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


Click here to add your own comments

Join in and write your own page! It's easy to do. How? Simply click here to return to Calculator Questions & Comments - 01.



Copyright © 2008-2015. All rights reserved. Solving-Math-Problems.com