The area of a rectangular athletic field is represented by the expression 12y^3+48y^2+24y square meters. Write an algebraic expression to represent one possible set of dimensions (in the sense "length times width" ) of the athletic field. Include correct units with your solution.
To find the dimensions of the rectangular athletic field, we can use the given expression for its area: 12y^3 + 48y^2 + 24y square meters.
Let's factorize the expression first:
12y^3 + 48y^2 + 24y = 12y(y^2 + 4y + 2)
Now, we have a product of two terms: 12y and (y^2 + 4y + 2).
To represent the dimensions of the athletic field, we need to find two factors whose product equals 12y. One factor will represent the length, and the other factor will represent the width.
For the remaining term (y^2 + 4y + 2), we can express it as a quadratic expression, as 2(y^2 + 2y + 1), which can be further factored as 2(y + 1)^2.
Therefore, the algebraic expression to represent one possible set of dimensions for the athletic field is:
12y * 2(y + 1)^2
This implies that the length is 12y meters and the width is 2(y + 1) meters.