SOLUTION: Hello, I'm not sure if this is the right section for this word problem. But I'm not sure what category it would fit into. So the problem is: You are designing a rectangular garden

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Question 1114119: Hello, I'm not sure if this is the right section for this word problem. But I'm not sure what category it would fit into. So the problem is: You are designing a rectangular garden and you have 56 feet of fencing. Is it possible to build a garden that is 10ft by 18ft? What would be the area? Can you build one that is 12ft by 16ft? What would be the area of that garden. What is the largest one that can be built?
I understand area. What I don't understand is what the problem is asking me. So when it says 10ft by 18ft, am I multiplying? Thank you so much! Any help is appreciated!
Found 3 solutions by josgarithmetic, greenestamps, ikleyn:
Answer by josgarithmetic(39623)   (Show Source): You can put this solution on YOUR website!
56 feet of fencing for PERIMETER

Will the rectangular garden be as much as or less than the 56 feet?


---------yes, true.

Could the dimensions be instead, 12 by 16 ?


----------yes, also true.

Perimeter in either case would be 56 feet, matching the amount of fence material.

AREA?
Area is width multiplied by length.
Answer by greenestamps(13203)   (Show Source): You can put this solution on YOUR website!


The perimeter -- twice length plus twice width -- is to be 56 feet, so length plus width should be 56/2=28 feet.

So the rectangular garden with a perimeter of 56 feet can be either 10ft by 18ft (10+18=28) or 12ft by 16ft (12+16=28).

What the question is asking for is the largest area you could get for a rectangular garden with a perimeter of 56 feet.

For the two examples of dimensions that the problem gives, you have
10*18 = 180 sq ft
12*16 = 192 sq ft

A general principle (and therefore a very useful one!) is that, for a fixed sum of two numbers, the maximum product is if the two numbers are the same.

So since the sum of length and width in this problem is 28, the maximum area is when the garden is 14ft by 14ft:
14*14 = 196 sq ft.
Answer by ikleyn(52832)   (Show Source): You can put this solution on YOUR website!
.
If the perimeter of a rectangle is given (as in your case the length of the fence of 56 feet),  then the rectangle which has 
MAXIMAL area is a square with the side length of    of the perimeter, i.e. of 14 ft in your case.


The maximal area then is   = 196 square feet.

See the lessons
    - HOW TO complete the square to find the minimum/maximum of a quadratic function
    - Briefly on finding the minimum/maximum of a quadratic function
    - HOW TO complete the square to find the vertex of a parabola
    - Briefly on finding the vertex of a parabola
    - A rectangle with a given perimeter which has the maximal area is a square (*)

The most relevant lesson is marked (*) in this list.

Also,  you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this textbook under the topic "Finding minimum/maximum of quadratic functions".


Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.


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