To ensure that the pool is a rectangle, the diagonals of the rectangle, [tex]\( AC \)[/tex] and [tex]\( BD \)[/tex], must be equal. This means we need to solve the equation [tex]\( AC = BD \)[/tex].
Given:
[tex]\[ AC = 10y + 4 \][/tex]
[tex]\[ BD = 13y - 8 \][/tex]
We set the expressions for [tex]\( AC \)[/tex] and [tex]\( BD \)[/tex] equal to each other:
[tex]\[ 10y + 4 = 13y - 8 \][/tex]
Now, solve for [tex]\( y \)[/tex]:
1. Subtract [tex]\( 10y \)[/tex] from both sides:
[tex]\[ 4 = 3y - 8 \][/tex]
2. Add 8 to both sides:
[tex]\[ 12 = 3y \][/tex]
3. Divide both sides by 3:
[tex]\[ y = 4 \][/tex]
Thus, the value of [tex]\( y \)[/tex] that ensures the pool is a rectangle is [tex]\( 4 \)[/tex].
