Final answer:
The dimensions of the rectangle of maximum area that can be formed using 3 m of wire are x = 3/4 m and y = 3/4 m.
Step-by-step explanation:
To find the dimensions of the rectangle of maximum area that can be formed using 3 m of wire, we need to use the fact that the perimeter of a rectangle is equal to twice the sum of its length and width.
Let's assume the width is x and the length is y. According to the given information, the perimeter of the rectangle is 3 m, so we have the equation 2(x + y) = 3.
To find the dimensions that maximize the area, we need to find the maximum value of the area function A = xy.
We can rewrite the equation for the perimeter as y = (3 - 2x)/2 and substitute it into the area function: A = x(3 - 2x)/2.
To find the maximum value of A, we can take the derivative of A with respect to x and set it equal to zero.
Differentiating A with respect to x, we get: dA/dx = (3 - 4x)/2. Setting this equal to zero and solving for x, we find
x = 3/4.
Substituting x = 3/4 back into the equation for y, we get y = (3 - 2(3/4))/2
= 3/4.
So, the dimensions of the rectangle of maximum area that can be formed using 3 m of wire are x = 3/4 m and y = 3/4 m.