To find the dimensions of a rectangle with a perimeter of 60 meters that maximizes its area, we can start with some basic formulas:
The perimeter (P) of a rectangle is given by the formula: P = 2(length + width). In this case, we know the perimeter is 60 meters, so we can write:
60 = 2(length + width)
Dividing both sides by 2 gives us:
30 = length + width
Next, we can express the width (w) in terms of the length (l):
w = 30 – l
The area (A) of a rectangle is given by the formula: A = length × width. Substituting for width, we get:
A = l × (30 – l)
This can be rewritten as:
A = 30l – l²
This equation is a quadratic function in standard form, A = -l² + 30l, which opens downwards (since the coefficient of l² is negative), indicating that it has a maximum point.
To find the maximum area, we can use the vertex formula for a parabola:
l = -b / 2a
Where a = -1 and b = 30 for our area equation:
l = -30 / (2 × -1) = 15 meters
Now, substituting back to find the width:
w = 30 – l = 30 – 15 = 15 meters
Thus, the dimensions of the rectangle that maximize the area, given a perimeter of 60 meters, are:
Length = 15 meters
Width = 15 meters
Interestingly, this means that the rectangle is actually a square, with equal sides of 15 meters each, providing the largest possible area for the given perimeter.