To determine the value of [tex]\( y \)[/tex] that ensures the pool [tex]\( ABCD \)[/tex] is a rectangle, we need to verify when diagonal lengths [tex]\( AC \)[/tex] and [tex]\( BD \)[/tex] are equal. For a rectangle, the diagonals are equal in length.
The diagonals of the pool are given by the equations:
[tex]\[ AC = 15y - 7 \][/tex]
[tex]\[ BD = 2y + 6 \][/tex]
To find the value of [tex]\( y \)[/tex] that makes these diagonals equal, we set the expressions for [tex]\( AC \)[/tex] and [tex]\( BD \)[/tex] equal to each other:
[tex]\[ 15y - 7 = 2y + 6 \][/tex]
Let's solve this equation step-by-step:
1. Subtract [tex]\( 2y \)[/tex] from both sides to consolidate the terms involving [tex]\( y \)[/tex] on one side:
[tex]\[ 15y - 2y - 7 = 6 \][/tex]
[tex]\[ 13y - 7 = 6 \][/tex]
2. Add [tex]\( 7 \)[/tex] to both sides to isolate the [tex]\( y \)[/tex]-term:
[tex]\[ 13y = 6 + 7 \][/tex]
[tex]\[ 13y = 13 \][/tex]
3. Divide both sides by 13 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{13}{13} \][/tex]
[tex]\[ y = 1 \][/tex]
Now let's check the given multiple choice options to see if this [tex]\( y \)[/tex]-value is present. The given options are:
[tex]\[
-13, -1, 13
\][/tex]
Since our calculated [tex]\( y \)[/tex]-value of 1 does not match any of the provided options, we find that none of the given values for [tex]\( y \)[/tex] satisfies the condition [tex]\( AC = BD \)[/tex] ensuring the pool is a rectangle.
Hence, the correct answer is:
[tex]\[
\boxed{\text{None of the given values for } y\ \text{ensures the pool is a rectangle.}}
\][/tex]
