A rectangle has sides whose lengths in units are represented by polynomials as shown, where x is a positive integer. Which expression represents the perimeter of the rectangle in units? (1 point)

Question

the rectangle is 3x - 1/2 in hight and 3/2 x + 3 in width.

A 9/2 x + 5/2
B 9x + 5
C 9/2 x + 7
D 9x + 7

Answer 1

The perimeter \( P \) of a rectangle is given by the formula:

\[ P = 2 \times (\text{height} + \text{width}) \]

Given the height of the rectangle is \( 3x - \frac{1}{2} \) and the width is \( \frac{3}{2} x + 3 \), we first need to add the height and the width:

\[ \text{height} + \text{width} = \left( 3x - \frac{1}{2} \right) + \left( \frac{3}{2} x + 3 \right) \]

Now combine the like terms:

1. Combine \( 3x \) and \( \frac{3}{2} x \): \[ 3x = \frac{6}{2} x \] \[ \frac{6}{2} x + \frac{3}{2} x = \frac{6 + 3}{2} x = \frac{9}{2} x \]

2. Combine \( -\frac{1}{2} \) and \( 3 \): \[ 3 = \frac{6}{2} \quad \Rightarrow \quad -\frac{1}{2} + \frac{6}{2} = \frac{5}{2} \]

So, we have:

\[ \text{height} + \text{width} = \frac{9}{2} x + \frac{5}{2} \]

Now we find the perimeter by multiplying this sum by 2:

\[ P = 2 \times \left( \frac{9}{2} x + \frac{5}{2} \right) \] \[ P = 2 \cdot \frac{9}{2} x + 2 \cdot \frac{5}{2} \] \[ P = 9x + 5 \]

Thus, the expression that represents the perimeter of the rectangle is:

B \( 9x + 5 \)