Given the sum of all the edges of the rectangular solid at the right is 76cm, the area of all of its faces is 228 cm^2, and its volume is 216 cm^3.

Find its height, width and length.
[hint- Let the height, width and length be the roots of a cubic polynomial x^3 + ax^2 + bx +c=0. How are the height, width and length related to a, b, and c? How are the above data related to a, b, and c?]

To find the height, width, and length of the rectangular solid, we need to solve the cubic polynomial equation x³ + ax² + bx + c = 0. The coefficients a, b, and c are related to the characteristics of the rectangular solid.

Let's determine how the data provided relates to the coefficients a, b, and c:

1. The sum of all the edges of the rectangular solid is 76 cm. Each edge of the rectangular solid contributes twice to the total sum since each edge has two ends. So, the sum of all the edges is given by the formula: 2(length + width + height) = 76. This equation provides one relationship between a, b, and c.

2. The area of all the faces of the rectangular solid is 228 cm². Each face has an area equal to the product of two adjacent edges. The total surface area of the rectangular solid can be calculated as: 2(length * width + width * height + height * length) = 228. This equation provides another relationship between a, b, and c.

3. The volume of the rectangular solid is 216 cm³. The volume of a rectangular solid is given by the formula: length * width * height = 216. This equation provides a third relationship between a, b, and c.

To find the height, width, and length, we need to solve these three equations simultaneously.