john 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 rectangle is length times width). What should the dimensions of the patio be, and show how the maximum area of the patio is calculated from the algebraic equation.
Show clearly the algebraic steps which prove your dimensions are the maximum area which can be obtained. Use the vertex formula to find the maximum area.
2006-11-25 14:01:48 · 6 answers · asked by howsurgrass 1 in Science & Mathematics ➔ Mathematics
Let the length be x ft and the width be y ft
2x + 2y = 300
ie x + y = 150
Thus y = 150 - x
Area = xy
=x(150 - x)
= 150x - x²
Vertex of this parabola occurs when x = -b/2a and at this point Area is maximum as the parabola is concave down. (a < 0)
ie x = (-150)/(-2)
= 75
When x = 75 y = 75 so it should be square with sides 75 ft
When x = 75, Area = 150*75 - 75² = 5625 sq ft
So the maximum area occurs when the length (and width) are 75 ft and it equals 5625 sq ft
2006-11-25 14:11:32 · answer #1 · answered by Wal C 6 · 0⤊ 0⤋
I don't know what the vertex formula is, but the shape of the patio is square. In this case, each side of the square would be 75 feet, so the area is 5,625 sq ft.
Why is the answer a square? If you use any other rectangle, you have added some amount (call it x feet) to one dimension (e.g., length) and subtracted the same amount from the other dimension (width). So the dimensions of the patio become (75 + x) by (75 - x).
The area of such a rectangle is 5,625 - x^2 sq ft. This area is maximized by setting x equal to 0. In other words, make the patio square.
2006-11-25 14:09:01 · answer #2 · answered by actuator 5 · 0⤊ 0⤋
For a given perimeter, a square has the largest possible area of a regular quadrilateral.
The patio should be 75 by 75 feet.
2006-11-25 14:14:51 · answer #3 · answered by discoganya 2 · 0⤊ 0⤋
You need to make the formula for the perimeter and the formula for the area. And then solve for the width or the length like (150 - w = L ), then put that formula into the original and solve for the other.
You basically just have to combine the area and perimeter formulas and solve.
This sounds more like junior high algebra than college.
2006-11-25 14:10:31 · answer #4 · answered by aroundthecorner_bumpme 2 · 0⤊ 0⤋
5,525 square feet
65 x 85 =
65 + 65 + 85 + 85 = 300 --- perimeter
if it is a rectangle
if it was 75 x 75 - 5,625 it would be a square
2006-11-25 14:09:23 · answer #5 · answered by goodcharacter 3 · 0⤊ 0⤋
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2016-12-10 16:07:09 · answer #6 · answered by Anonymous · 0⤊ 0⤋