Question

A rectangle has sides whose lengths in units are represented by polynomials as shown where x is a positive integer
3/2x+3
3x-1/2
Which expression represents the perimeter of the rectangle in units
A. 9/2x+5/2

Answer 1

To find the perimeter \( P \) of a rectangle given by the side lengths represented by two polynomials, we can use the formula for perimeter:

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

In this case, the lengths of the sides are represented as:

- Length \( L = \frac{3}{2}x + 3 \)
- Width \( W = 3x - \frac{1}{2} \)

Now let's find \( L + W \):

\[
L + W = \left( \frac{3}{2}x + 3 \right) + \left( 3x - \frac{1}{2} \right)
\]

Combining 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:

\[
L + W = \frac{9}{2}x + \frac{5}{2}
\]

Now, we need to calculate the perimeter:

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

Distributing the 2:

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

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

\[
P = 9x + 5
\]

None of the options provided match this, but it appears the expression listed as option A, \( \frac{9}{2}x + \frac{5}{2} \), is not correct as it does not correctly represent the perimeter. The correct perimeter expression is \( 9x + 5 \).