Explanation:
Let L and W be the length and width of the rectangle, respectively.
We know that the perimeter, P, of a rectangle is given by:
P = 2L + 2W
In this case, P = 16 cm, so we have:
16 = 2L + 2W
Simplifying, we get:
8 = L + W
To find the greatest area of the rectangle, we need to maximize the product of L and W, which is the formula for the area, A:
A = L * W
We can solve for one variable in terms of the other using the equation we found earlier:
L = 8 - W
Substituting this into the formula for the area, we get:
A = (8 - W) * W
Expanding and simplifying, we get:
A = 8W - W^2
To find the maximum value of A, we can use calculus or complete the square. Completing the square, we get:
A = -(W - 4)^2 + 16
Since the square of a real number is always nonnegative, the maximum value of A occurs when (W - 4)^2 = 0, which is when W = 4.
Substituting this value back into the equation for the perimeter, we get:
8 = L + 4
L = 4
Therefore, the rectangle with a perimeter of 16 cm and the greatest area is a square with sides of length 4 cm, and its area is:
A = L * W = 4 * 4 = 16 square centimeters.