The number of wrenches that can be produced at a given price can be predicted by the formula $s = \sqrt { 70 x }$ , where $s$ is the supply (in thousands)and $X$ is the price (in dollars). If the demand $d$ for the wrenches can be predicted by the formula $d = \sqrt { 600 - 5 x ^ { 2 } }$ , find the equilibrium price.

Question

Quizplus · Accepted Answer

At equilibrium, supply equals demand. Therefore, we need to set the two formulas equal to each other and solve for the price:

$$\sqrt{70x} = \sqrt{600 - 5x^2}$$

Squaring both sides, we get:

$$70x = 600 - 5x^2$$

Simplifying and rearranging, we get:

$$5x^2 + 70x - 600 = 0$$

Dividing both sides by 5, we get:

$$x^2 + 14x - 120 = 0$$

Factoring, we get:

$$(x+20)(x-6) = 0$$

Therefore, the possible values for the price are $-20$ and $6$. Since we are told that the price is in dollars, we can eliminate the negative solution, leaving us with:

$$x = 6$$

Therefore, the equilibrium price is $\boxed{\textbf{(C)}\ 6}$.

The Number of Wrenches That Can Be Produced at a Given