To solve this problem, we must determine the correct expression for the side length of a square given its perimeter. The perimeter of a square is the total length around its four sides, which is calculated as:
[tex]\[ \text{Perimeter} = 4 \times \text{side length} \][/tex]
The problem states that the perimeter of the square is [tex]\( 80 - 64y \)[/tex] units. Therefore:
[tex]\[ 80 - 64y = 4 \times \text{side length} \][/tex]
To find the side length, we need to isolate it by dividing both sides of the equation by 4:
[tex]\[ \text{side length} = \frac{80 - 64y}{4} \][/tex]
Simplifying the right side:
[tex]\[ \text{side length} = \frac{80}{4} - \frac{64y}{4} \][/tex]
[tex]\[ \text{side length} = 20 - 16y \][/tex]
Now we compare this expression to the given options:
1. [tex]\( 16(5-4y) \)[/tex]: Expanding this expression:
[tex]\[ 16 \times (5 - 4y) = 16 \times 5 - 16 \times 4y \][/tex]
[tex]\[ 16 \times (5 - 4y) = 80 - 64y \][/tex]
Here, we see that [tex]\( 16(5-4y) \)[/tex] actually matches the given perimeter, not the side length. Each side does not measure [tex]\( 5-4y \)[/tex] units using this option.
2. [tex]\( 16y(5-4) \)[/tex]: Expanding this expression:
[tex]\[ 16y \times (5 - 4) = 16y \times 1 = 16y \][/tex]
This option, 16y, does not match any part of the original relation of the side length and does not represent the side length.
3. [tex]\( 4(20-16y) \)[/tex]: Expanding this expression:
[tex]\[ 4 \times (20 - 16y) = 4 \times 20 - 4 \times 16y \][/tex]
[tex]\[ 4 \times (20 - 16y) = 80 - 64y \][/tex]
This option, 4(20-16y), actually matches our derived side length. Each side measures [tex]\( 20 - 16y \)[/tex] units using this option.
4. [tex]\( 4y(20-16) \)[/tex]: Expanding this expression:
[tex]\[ 4y \times (20 - 16) = 4y \times 4 = 16y \][/tex]
This option, 16y, does not match any part of the original derived side length and does not represent the side length.
Thus, the correct expression that shows the side length of one side of the square is [tex]\( 4(20-16y) \)[/tex]. Therefore, the side length of the square is given by:
[tex]\[ 20 - 16y \][/tex]
Each side measures:
[tex]\[ 20 - 16y \][/tex] units.
