Rectangular patio

[pooh27]

mark has 300 feet of lumber to frame a rectangular patio (the perimeter of a rectangle is 2 times length plus 2 times width). He wants to maximize the area of his patio (area of a retangle is lenght times width). What should the dimensions of the patio be? Show how the maximum area of the patio is calculated from the algebraic equations.


Any kind of help would be greatful.

 

[stapel]

A good start might be to pick variables for the length and the width, and to translate the given relationships into equations.

Eliz.

 

[pooh27]

A good start might be to pick variables for the length and the width....

How would I pick the variables for the length and width. I am not good at word problems.

 

[morson]

How would I pick the variables for the length and width. I am not good at word problems.

Let the width be w. Let the length be l.

The first sentence directly tells you 2w + 2l = 300, ie: w + l = 150
The second sentence tells you A = wl

A geometrical interpretation could also help you. You have two side lengths of a rectangle. To achieve a maximum area, they should be the same length. Ie: the rectangle, with a given perimeter, with the largest possible area is a square. This exercise just proves that result.

 

[stapel]

How would I pick the variables for the length and width.

Have you never worked with any geometric stuff at all? (Anybody who has is familiar with the standard variables "L" and "w" for "length" and "width", respectively, is why I ask.)

We can find lesson links to help you learn the necessary background information, if you would kindly please specify which topics you would like.

Thank you!

Eliz.

P.S. You posted this to "Intermediate/Advanced Algebra", so you've had earlier algebra couses. Those classes usually (always?) cover this beginning material at length, since you can't very well do algebra without variables. What curriculum are you using, that your previous classes didn't use variables? (I like to familiarize myself with new trends in education, and this is something I've never heard of.) Thank you!

 

[pooh27]

Eliz,

I am in College Algebra. It has been 8 years since I have had to do any Algebra. It is that it is taking me awhile to get back into the swing of things. I apperacite the help that you have provided me with. Thank you, Candace Aldridge

 

[stapel]

It has been 8 years since I have had to do any Algebra.

And they didn't administer a placement test, so you could start in an appropriate class? Ouch! You have my sympathies! :shock:

To help make up for what they didn't give you, try here to learn about variables, here to learn about geometric formulas, and here to learn how to convert English statements into mathematical expressions and equations.

Good luck!

Eliz.

 

[TchrWill]

mark has 300 feet of lumber to frame a rectangular patio (the perimeter of a rectangle is 2 times length plus 2 times width). He wants to maximize the area of his patio (area of a retangle is lenght times width). What should the dimensions of the patio be? Show how the maximum area of the patio is calculated from the algebraic equations.

You do not say whether the house itself will be one edge of the restngular patio.

Considering all rectangles with a given perimeter, the square encloses the greatest area.

Proof: Consider a square of dimensions x by x, the area of which is x^2. Adjusting the dimensions by adding a to one side and subtracting a from the other side results in an area of (x + a)(x - a) = x^2 - a^2. Thus, however small the dimension "a" is, the area of the modified rectangle is always less than the square of area x^2.

Considering all rectangles with a given perimeter, the square encloses the greatest area.
Proof: Consider a square of dimensions x by x, the area of which is x^2. Adjusting the dimensions by adding a to one side and subtracting a from the other side results in an area of (x + a)(x - a) = x^2 - a^2. Thus, however small the dimension "a" is, the area of the modified rectangle is always less than the square of area x^2.

Considering all rectangles with the same area, the square results in the smallest perimeter for a given area.


Considering all rectangles with a given perimeter, one side being another straight boundry, the 3 sided
rectangle enclosing the greatest area has a length to width ratio of 2:1