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Among all of the rectangles with a perimeter of 20ft, find the dimensions of the one with the largest area.?
Like the question says.
(Pre-Calculus)
1 Answer
- MoonRoseLv 71 decade ago
Let l be the length and w be the width. The perimeter of a rectangle is 2l + 2w. Write the length in terms of the width.
2l + 2w = 20
l + w = 10
l = 10 - w
The area of a rectangle is lw. Now that you know the length is equivalent to 10 - w, you can write the area as w(10 - w) using substitution. The width that will create the greatest area is -b/(2a), from the quadratic ax^2 + bx + c.
w(10 - w)
10w - w^2
-w^2 + 10w ===> a = -1, b = 10, c = 0
w = -b/(2a)
w = -10/(2*-1)
w = -10/-2
w = 5
The width that will create the largest area is 5 feet, which makes the length = 10 - w = 10 - 5 = 5 feet. Therefore the largest area of a rectangle with a perimeter of 20 is 5(5) = 25 square feet.
ANSWER: 5 feet by 5 feet
