To determine the expression used to find the perimeter of Zohar's rectangle and to calculate the perimeter when [tex]\( x = 4 \)[/tex], follow these steps:
1. Identify the Length and Width of the Rectangle:
- The length of the rectangle is given by [tex]\( 5x - 2 \)[/tex].
- The width of the rectangle is given by [tex]\( 3x + 1 \)[/tex].
2. Perimeter of a Rectangle:
- The perimeter ([tex]\( P \)[/tex]) of a rectangle is calculated using the formula [tex]\( P = 2 \times (\text{length} + \text{width}) \)[/tex].
3. Substitute the Expressions for the Length and Width:
- Substitute [tex]\( 5x - 2 \)[/tex] for the length and [tex]\( 3x + 1 \)[/tex] for the width into the perimeter formula:
[tex]\[
P = 2 \times ((5x - 2) + (3x + 1))
\][/tex]
4. Simplify the Expression inside the Parentheses:
- Combine like terms inside the parentheses:
[tex]\[
P = 2 \times (5x - 2 + 3x + 1)
\][/tex]
[tex]\[
P = 2 \times (8x - 1)
\][/tex]
5. Expression for the Perimeter:
- The expression used to find the perimeter of the rectangle is [tex]\( 2 \times (5x - 2) + 2 \times (3x + 1) \)[/tex].
6. Calculate the Perimeter for [tex]\( x = 4 \)[/tex]:
- When [tex]\( x = 4 \)[/tex], first calculate the length and width:
[tex]\[
\text{Length} = 5 \times 4 - 2 = 20 - 2 = 18
\][/tex]
[tex]\[
\text{Width} = 3 \times 4 + 1 = 12 + 1 = 13
\][/tex]
- Then, substitute these values into the perimeter formula:
[tex]\[
P = 2 \times (18 + 13)
\][/tex]
[tex]\[
P = 2 \times 31 = 62
\][/tex]
Hence, the expression used to find the perimeter of the rectangle is [tex]\( 2(5x - 2) + 2(3x + 1) \)[/tex], and the perimeter when [tex]\( x = 4 \)[/tex] is 62 centimeters. Therefore, the correct choice is:
[tex]\[
2(5x - 2) + 2(3x + 1); 62 \text{ centimeters}
\][/tex]
