Julissa wants to join her local gym. The membership cost is \( \$ 282.90 \) for 6 months and \( \$ 448.80 \) for one year. Which of money, \( y \), she will spend on her gym membership for \( x \) months? \( \begin{array}{l}y=83.82 x+33.18 \\ y=117 x+27.65 \\ y=33.18 x+83.82 \\ y=27.65 x+117\end{array} \)

To determine the relationship between the number of months Julissa spends on her gym membership and the total cost, let's first analyze the costs. For 6 months, the cost is \( \$ 282.90 \), which gives us a rate of about \( \$ 47.15 \) per month. For a year (12 months), the total cost is \( \$ 448.80 \), which translates to approximately \( \$ 37.40 \) per month. To create the total cost equation, we can segment it based on the time frame. For \( x \) months, if she opts for the 6-month plan and continues afterward, then the cost for \( x \) months where \( x \leq 6 \) is simply the \( \$ 47.15 \) rate. If \( x \) exceeds 6 months, then she modifies her approach, which could lead towards the 12-month rate, usually resulting in a more cost-effective option. When looking at the options presented: none represent either a simplistic linear relationship based solely on the membership plan, as numbers seem off. Remember, a linear equation for this would be structured around the discounted monthly rate for extended use. The answer to the problem lies in setting the equation for each segment from \( x = 1 \) to \( 6 \) and \( x = 7 \) to \( 12 \) and adjusting accordingly—working through careful math ensures she gets the most out of her dollars! The approach would require some refining of the given equations to align with the gym's pricing structure accurately. Make sure to check the calculations again.