Question

the perimeter of the rectangle below is 116 units. find the value of z.
rectangle with length 3z and width 2z + 3
z = \boxed{}

Explanation:

Step1: Recall the perimeter formula for a rectangle

The perimeter \( P \) of a rectangle is given by \( P = 2\times(\text{length} + \text{width}) \). Here, the length is \( 3z \) and the width is \( 2z + 3 \), and the perimeter \( P = 116 \). So we substitute into the formula:

$$ 116 = 2\times(3z + (2z + 3)) $$

Step2: Simplify the equation

First, simplify the expression inside the parentheses: \( 3z + 2z + 3 = 5z + 3 \). Then the equation becomes:

$$ 116 = 2\times(5z + 3) $$

Distribute the 2 on the right - hand side:

$$ 116 = 10z + 6 $$

Step3: Solve for \( z \)

Subtract 6 from both sides of the equation:

$$ 116 - 6 = 10z + 6 - 6 $$

$$ 110 = 10z $$

Divide both sides by 10:

$$ z=\frac{110}{10}=11 $$

Answer:

\( z = 11 \)