To determine the value of [tex]\( y \)[/tex] which ensures that the pool is a rectangle, follow these steps:
1. Understand the Given Expressions:
- The lengths of the diagonals [tex]\( AC \)[/tex] and [tex]\( BD \)[/tex] are given by the expressions:
[tex]\[
AC = 5y + 6
\][/tex]
[tex]\[
BD = 8y - 3
\][/tex]
2. Rectangle Property:
- For the pool [tex]\( ABCD \)[/tex] to be a rectangle, its diagonals must be equal in length. Therefore, we set the expressions for [tex]\( AC \)[/tex] and [tex]\( BD \)[/tex] equal to each other:
[tex]\[
5y + 6 = 8y - 3
\][/tex]
3. Solve the Equation:
- To find [tex]\( y \)[/tex], solve the equation [tex]\( 5y + 6 = 8y - 3 \)[/tex]:
[tex]\[
5y + 6 = 8y - 3
\][/tex]
[tex]\[
6 + 3 = 8y - 5y
\][/tex]
[tex]\[
9 = 3y
\][/tex]
[tex]\[
y = \frac{9}{3} = 3
\][/tex]
4. Verify the Solution:
- To ensure that [tex]\( y = 3 \)[/tex] is an appropriate answer, we can check the values of the diagonals when [tex]\( y = 3 \)[/tex]:
[tex]\[
AC = 5(3) + 6 = 15 + 6 = 21
\][/tex]
[tex]\[
BD = 8(3) - 3 = 24 - 3 = 21
\][/tex]
- Since both [tex]\( AC \)[/tex] and [tex]\( BD \)[/tex] equal 21 when [tex]\( y = 3 \)[/tex], the condition that the diagonals are equal is satisfied.
Therefore, the value of [tex]\( y \)[/tex] that ensures the pool is a rectangle is [tex]\(\mathbf{3}\)[/tex].
