Question

A rectangle has sides whose lengths in units are represented by polynomials as shown, where x is a positive integer.

Width: 3/2x + 3 height: 3x - 1/2

Which expression represents the perimeter of the rectangle in units?

9/2x+5/2
9 halves x plus 5 halves

9x+5
9 x plus 5

9/2x+7/2
9 halves x plus 7 halves

9x+7

Answer 1

To find the perimeter of a rectangle, we use the formula:

\[
\text{Perimeter} = 2 \times (\text{Width} + \text{Height})
\]

Given that the width is \( \frac{3}{2}x + 3 \) and the height is \( 3x - \frac{1}{2} \), let's first calculate \( \text{Width} + \text{Height} \):

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

Now, combine the terms:

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

2. Combine the constant terms:
\[
3 - \frac{1}{2} = \frac{6}{2} - \frac{1}{2} = \frac{5}{2}
\]

So, we have:

\[
\text{Width} + \text{Height} = \frac{9}{2}x + \frac{5}{2}
\]

Now substitute this back into the perimeter formula:

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

Distribute the \( 2 \):

\[
\text{Perimeter} = 2 \times \frac{9}{2}x + 2 \times \frac{5}{2}
\]

This simplifies to:

\[
\text{Perimeter} = 9x + 5
\]

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

\[
\boxed{9x + 5}
\]