\( \left. \begin{array} { l } { p = d ( x ) = 144 - x ^ { 2 } } \\ { p = s ( x ) = 48 + 0.5 x ^ { 2 } } \end{array} \right. \)

To find the intersection points of the two functions, set them equal to each other: \[ 144 - x^2 = 48 + 0.5x^2 \] Rearranging gives: \[ 144 - 48 = x^2 + 0.5x^2 \] This simplifies to: \[ 96 = 1.5x^2 \] Dividing both sides by 1.5 results in: \[ x^2 = 64 \] Therefore, \( x = 8 \) or \( x = -8 \). Now, substituting these x-values back into either equation gives you the corresponding prices. For \( x = 8 \): \[ p = 144 - 8^2 = 144 - 64 = 80 \] And for \( x = -8 \): \[ p = 144 - (-8)^2 = 80 \] Thus, the two points of intersection (supply meets demand) are at \( (8, 80) \) and \( (-8, 80) \). Now, let’s talk about the real-world application: understanding these curves helps businesses determine the supply and demand balance. If the price is set too high, consumers won't buy, leading to excess supply. Conversely, if prices are too low, there won't be enough supply to meet demand. This principle is crucial for any successful market strategy – think of it as the dance between buyers and sellers, trying to find that sweet spot! Don’t forget to watch out for common mistakes! One of the biggest is misinterpreting the intersections and thinking they represent feasible real-world scenarios. In this case, while \( x = -8 \) mathematically works, it doesn’t make sense in a practical context since negative quantities of products typically aren’t feasible. Always focus on interpreting your results in the context of the situation!