To allow for more dancers the country club can increase the dimensions of the dance floor in 5 foot increments in either direction ; however the committee would like to maintain a square shape. Let n represent the number of 5'increases to the length pf each side of the dance floor. write an expression in terms of n for the additional square feet of dance floor and then use algebraic identities to find a diffrent but equivalent expression for the additional area.

total area= (l+5n)^2

original area= l^2

new area= total area-original= 10*L*n+ 25n^2

check that with a drawing.

To write an expression in terms of n for the additional square feet of dance floor, we first need to determine how the length of each side of the dance floor relates to the number of 5-foot increments, n.

Since each 5-foot increment increases both the length and width of the dance floor, if we start with an initial side length of x, then the length of each side after n increments will be x + 5n.

Now, to determine the additional square feet of dance floor, we need to find the difference between the area after the increments and the initial area.

The initial area of the dance floor is given by x * x = x^2.

After n increments, the length of each side is x + 5n, so the area of the dance floor is (x + 5n) * (x + 5n) = (x + 5n)^2.

Therefore, the additional square feet of dance floor can be expressed as (x + 5n)^2 - x^2.

To find a different but equivalent expression for the additional area, let's expand (x + 5n)^2 using the algebraic identity (a + b)^2 = a^2 + 2ab + b^2:

(x + 5n)^2 = x^2 + 2x * 5n + (5n)^2
= x^2 + 10xn + 25n^2.

Substituting this back into the expression for the additional square feet of dance floor, we get:

(x + 5n)^2 - x^2 = (x^2 + 10xn + 25n^2) - x^2
= 10xn + 25n^2.

Therefore, a different but equivalent expression for the additional area is 10xn + 25n^2.