Answer :
The dimensions of the rectangle with the largest area are x = 54.5 units and y = 54.5 units.
To find the rectangle with the largest area among those with a perimeter of 218, we can follow these steps:
1. **Let the dimensions of the rectangle be x (length) and y (width).**
2. **Express the perimeter in terms of x and y:** Perimeter = 2x + 2y = 218.
3. **Solve for one variable in terms of the other:** From the perimeter equation, y = (218 - 2x) / 2.
4. **Express the area as a function of one variable:** Area = xy = x * ((218 - 2x) / 2).
5. **Simplify the area function:** Area = x(109 - x).
6. **Find the maximum value of the area function:** This can be done by analyzing the graph of the area function or using calculus. The parabola opens downwards, reaching its maximum value at the vertex.
7. **Find the value of x at the vertex:** Vertex x-coordinate = -b / 2a, where a and b are the coefficients of the quadratic term (x^2) and linear term (-x), respectively. In this case, a = -1 and b = 109, so the vertex x-coordinate is -109 / (2 * -1) = 54.5.
8. **Substitute the vertex x-coordinate back into the equation for y to find the corresponding y-value:** y = (218 - 2 * 54.5) / 2 = 54.5.
9. **Therefore, the dimensions of the rectangle with the largest area are x = 54.5 units and y = 54.5 units.**
