$$ p=d(x)=144-x^{2} $$ $$ p=s(x)=48+0.5 x^{2} $$ 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

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To graph the demand function $$ d(x) = 144 - x^2 $$ and the supply function $$ s(x) = 48 + 0.5x^2 $$, we need to determine the appropriate domain for each function.

For the demand function $$ d(x) = 144 - x^2 $$, the domain is all real numbers since the function is defined for any value of $$ x $$.

For the supply function $$ s(x) = 48 + 0.5x^2 $$, the domain is also all real numbers since the function is defined for any value of $$ x $$.

To find the market equilibrium, we need to set the demand function equal to the supply function and solve for $$ x $$:
$$ 144 - x^2 = 48 + 0.5x^2 $$

Solving this equation will give us the value of $$ x $$ at the market equilibrium.

Let's solve the equation to find the market equilibrium point.
Solve the quadratic equation by following steps:
- step0: Solve using square roots:
  $$144-x^{2}=48+0.5x^{2}$$
- step1: Convert the decimal into a fraction:
  $$144-x^{2}=48+\frac{1}{2}x^{2}$$
- step2: Add or subtract both sides:
  $$-\frac{3}{2}x^{2}=-96$$
- step3: Divide both sides:
  $$\frac{-\frac{3}{2}x^{2}}{-\frac{3}{2}}=\frac{-96}{-\frac{3}{2}}$$
- step4: Divide the numbers:
  $$x^{2}=64$$
- step5: Simplify the expression:
  $$x=\pm \sqrt{64}$$
- step6: Simplify the expression:
  $$x=\pm 8$$
- step7: Separate into possible cases:
  $$\begin{align}&x=8\&x=-8\end{align}$$
- step8: Rewrite:
  $$x_{1}=-8,x_{2}=8$$
The market equilibrium point is at $$ x = 8 $$ or $$ x = -8 $$. Since the number of Super Titan tires cannot be negative, the market equilibrium point is at $$ x = 8 $$.

Therefore, the market equilibrium point is at $$ x = 8 $$ or 8,000 Super Titan tires.

Now, let's calculate the price at the market equilibrium point by substituting $$ x = 8 $$ into either the demand function or the supply function.
Evaluate the expression by following steps:
- step0: Evaluate:
  $$144-x^{2}$$
- step1: Substitute:
  $$144-8^{2}$$
- step2: Evaluate the power:
  $$144-64$$
- step3: Subtract the numbers:
  $$80$$
The price at the market equilibrium point is $80.

Therefore, the market equilibrium point is at $$ x = 8 $$ or 8,000 Super Titan tires, and the price is $80.

Now, let's determine a reasonable domain for the demand function. Since the number of Super Titan tires cannot be negative, the domain for the demand function is $$ x \geq 0 $$.
The market equilibrium is at 8,000 Super Titan tires with a price of $80. The domain for the demand function is $$ x \geq 0 $$.

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