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Show that the rectangle of maximum area for given perimeter P is always a square?
Suppose there exists some rectangle whose center is the point (x,y) = (0,0) with a given perimeter P and that we'd like to find the dimensions which maximize its area. The method we will use is called Lagrange Multipliers.
Then f(x,y) = P = 4x + 4y and g(x,y) A = xy.
(fx, fy) = L(gx, gy) (for some constant L which exists on the set of real numbers)
(4, 4) = L(y,x)
(4, 4) = (Ly, Lx)
But then Lx = 4 = Ly which yields Lx = Ly and x = y. Therefore when the area is maximized, the x component equals the y component. Furthermore, a rectangle centered at (0,0) whose corner falls on some point (x,y) at which x = y is a square.
What else can I help you with?
Can a rectangle have the same perimeter but different area explain?
No, any shape with four sides and same perimeter will always be a square.
The area of a rectangle is 100 inches the perimeter is 40 a second rectangle has the same area but a different perimeter. Is the second rectangle a square?
the area of a rectangleis 100 square inches. The perimeter of the rectangle is 40 inches. A second rectangle has the same area but a different perimeter. Is the secind rectangle a square? Explain why or why not.
The area of a rectangle is 100 square inches The perimeter of the rectangle is 40 inches A second rectangle has the same area but a different perimeter Is the second rectangle a square?
yes
What is the maximum area of a rectangle that can be made with a perimeter of 68 inches?
The maximum area with straight sides and a given perimeter is a square.The sides of the square are (68/4) = 17 inches.The area is (17 x 17) = 289 square inches
Is it sometimes always or never true that the perimeter of a rectangle is numerically greater than its area?
Sometimes. Experiment with a small square and with a large square (though any shape rectangle will do). A square of 4 x 4 has a perimeter of 16, and an area of 16. A smaller square has more perimeter than area. A larger square has more area than perimeter.
What is the maximum area for a rectangle whose perimeter is 136?
The maximum area for a rectangle of fixed perimeter is that of the square that can be formed with the given perimeter. 136/4 = 34, so that the side of such a square will be 34 and its area 342 = 1156.
What is the maximum area of a rectangle with a perimeter of 130?
130/4 (4 sides to a rectangle)= 32.5 32.5*32.5=1065.25 square meters (because the largest area of a rectangle is always a ^ ^ square). length width
What is the maximum area in square units of a rectangle with a perimeter of 128 units?
1024
What is the maximum area for a rectangle with a perimeter that is 88?
A square of side 22 has an area of 484. Rectangle 23 x 21 has an area of 483...
Can a rectangle have the same perimeter but different area explain?
No, any shape with four sides and same perimeter will always be a square.
The area of a rectangle is 100 inches the perimeter is 40 a second rectangle has the same area but a different perimeter. Is the second rectangle a square?
the area of a rectangleis 100 square inches. The perimeter of the rectangle is 40 inches. A second rectangle has the same area but a different perimeter. Is the secind rectangle a square? Explain why or why not.
What is the maximum area of a rectangular garden with a perimeter of 48 feet?
The maximum area is attained when the rectangle is, in fact, a square. Since the perimeter = 48 feet, the maximum length for a square = 48/4 = 12 feet. So max area = 122 = 144 square feet.
The area of a rectangle is 100 square inches The perimeter of the rectangle is 40 inches A second rectangle has the same area but a different perimeter Is the second rectangle a square?
yes
If you have an area of 100 square meters what is the perimeter?
40 meters.
What is the formula for the perimeter of a square and rectangle?
The perimeter of any rectangle is [ 2 x (length + width) ]. Since the length and width of a square are equal, the perimeter of a square is also [ 2 x (side + side) ] = (4 x side).
What is the maximum area of a rectangle that can be made with a perimeter of 68 inches?
The maximum area with straight sides and a given perimeter is a square.The sides of the square are (68/4) = 17 inches.The area is (17 x 17) = 289 square inches
Is it sometimes always or never true that the perimeter of a rectangle is numerically greater than its area?
Sometimes. Experiment with a small square and with a large square (though any shape rectangle will do). A square of 4 x 4 has a perimeter of 16, and an area of 16. A smaller square has more perimeter than area. A larger square has more area than perimeter.
