A rectangle has perimeter equal to 28, what is the maximum possible area of this rectangle.
max area is obtained with a square.
In this case, 7x7 = 49
To find the maximum possible area of a rectangle, we need to consider that the perimeter of a rectangle can be calculated using the formula P = 2l + 2w, where l is the length and w is the width.
Given that the perimeter of the rectangle is 28, we have the equation 2l + 2w = 28.
We want to find the maximum area, so we need to consider that the length and width of a rectangle should be as close to each other as possible.
To maximize the area, let's assume the width is equal to x. This means the length of the rectangle would also be x. Now we have:
2x + 2x = 28
4x = 28
x = 7
So the width (w) and length (l) are both 7.
Now, we can calculate the maximum possible area:
Area = length x width
Area = 7 x 7
Area = 49
Therefore, the maximum possible area of the rectangle with a perimeter of 28 is 49 square units.
To find the maximum possible area of a rectangle with a given perimeter, we need to use the concept of optimization. Let's start by analyzing the properties of a rectangle.
A rectangle has two pairs of equal sides. Let's call the length of each pair of equal sides "x" and "y." The perimeter of a rectangle is given by the formula:
Perimeter = 2x + 2y
In our case, the perimeter is equal to 28, so we have the equation:
28 = 2x + 2y
Now, we need to express one variable in terms of the other to solve the equation. Let's rearrange the equation to isolate y:
2y = 28 - 2x
y = 14 - x
Next, we need to express the area of the rectangle in terms of only one variable. The area of a rectangle is given by the formula:
Area = length * width
In this case, the length is x, and the width is y (or 14-x). Therefore, the area is:
Area = x * (14 - x)
Area = 14x - x^2
Now, we have the area in terms of x only. To find the maximum area, we can take the derivative of the area function with respect to x and set it equal to zero. This will give us the critical points where the area is maximized.
d(Area)/dx = 14 - 2x
Setting this derivative equal to zero:
14 - 2x = 0
2x = 14
x = 7
Now, we have x = 7 as a critical point. To determine whether it corresponds to a maximum or minimum, we can take the second derivative of the area function.
d^2(Area)/dx^2 = -2
Since the second derivative is negative, we conclude that x = 7 corresponds to a maximum area.
To find the maximum area, we substitute this value of x back into the area equation:
Area = 14x - x^2
Area = 14(7) - (7)^2
Area = 98 - 49
Area = 49
Therefore, the maximum possible area of a rectangle with a perimeter of 28 is 49 square units.