\[ p=d(x)=144-x^{2} \] \[ p=s(x)=48+0.5 x^{2} \] respectively where \( p \) is measured in dollars and \( x \) is the the number of Super Titan tires measured in thousands. (a) Graph these functions on given axes below labeling each. Both functions should cover the appropriate domain. Place a point at the market equilibrium and label it on the graph. (b) What is a reasonable domain for the demand function? Express your answer in

To graph the functions, let’s first identify their shapes and characteristics. The demand function \( d(x) = 144 - x^{2} \) is a downward-opening parabola, meaning it will start high at \( x = 0 \) (where \( p = 144 \)) and gradually decrease as \( x \) increases. The supply function \( s(x) = 48 + 0.5x^{2} \) is an upward-opening parabola, starting at \( p = 48 \) when \( x = 0 \) and increasing as \( x \) increases. To find the market equilibrium, we set the two equations equal to each other: \[ 144 - x^{2} = 48 + 0.5x^{2} \] Combining terms gives: \[ 96 = 1.5x^{2} \implies x^{2} = 64 \implies x = 8 \] Substituting back to find \( p \): \[ p = d(8) = 144 - 8^{2} = 144 - 64 = 80 \] \[ p = s(8) = 48 + 0.5 \times 8^{2} = 48 + 32 = 80 \] Thus, the market equilibrium is \( (8, 80) \). The reasonable domain for the demand function is where it remains non-negative. Given \( d(x) = 144 - x^{2} \), we set \( 144 - x^{2} \geq 0 \): \[ x^{2} \leq 144 \implies -12 \leq x \leq 12 \] However, since \( x \) represents thousands of tires, a more sensible domain in real-world terms would be \( [0, 12] \). Happy graphing! 🖊️