If you have 300 feet of lumber to frame a rectangular patio, what should the dimensions of the patio be?

Question

Show how the maximum area of the patio is calculated using algebraic equation.
*perimeter is 2X length + 2X length
*area is length X width
*use the vertex formula to find the maximum area.

Answer 1

perimeter formula for a rectangular

P = 2L + 2W

Let

100 = length

50 = Width

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P = 2(100) + 2(50)

P = 200 + 100

P = 300 Feet

The length is 100 feet

The width is 50 feet

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Area of a rectangular

A + L x W

A = 100 x 50

A = 5000 square feet

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Answer 2

What I think you are getting at is to say if length = x then width is 150 - x using your first starred line. Therefore area A = x(150 - x) by your second starred line. I don't know what you mean by vertex formula but you can find when this is a maximum by several means including differentiation if you know how to do that. That should be enough to get you started. Try to do it yourself before looking at anyone's detailed answer left here. (By the way the correct answer for the maximum area is more than 5000 sq. ft but less than 6000 sq. ft.)
All of this assumes that the patio is going to be surrounded on all four sides. If it is going to be surrounded on three sides then you will have to adapt the above method.

Answer 3

relative to what you want the thing to look like, probably 10 X 30
is your best bet, of course there are many variations --

Assuming 300 is in board ft, that is 1 foot wide by 300 ft long
or and area of 300 sq ft area. 300ft=L X W or, 10 X 30, or 5 X 60,
or 15 X 20.

Max area is gonna be 300 sq ft no matter what the rectangular
configuation, if you use all the lumber.