(1 point) A company rurs food service concessions for sporting events trroughout the country. Their marketing research departmert chose a perticuitr footbell stadium to test martat a now fumbo hot dog. It was found that the damund for the now hot dog is given approximatey by $$ p=4-\ln (x), \quad 5 \leq x \leq 500 $$ where $$ x $$ is the number of hot dogs (in thousanda) that can be sold during one game at a price of $$ p $$ dollars. It the compary pays 1 dollar for each hot dog. Now should the hot dogs be priced to maximize the proftit per game? Price = $$ \square $$

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Let $$ x $$ represent the number of hot dogs sold in thousands, and let $$ p $$ be the price in dollars. The demand function is given by

$$
p = 4 - \ln(x)
$$

The cost per hot dog is 1 dollar. In terms of thousands, the revenue per game is

$$
\text{Revenue} = p \cdot (1000x)
$$

and the cost is

$$
\text{Cost} = 1 \cdot (1000x)
$$

The profit per game (ignoring the constant factor 1000 that does not affect the optimization) is

$$
P(x) = x \cdot (p - 1)
$$

Substitute the expression for $$ p $$:

$$
P(x) = x \cdot \left[ \left(4 - \ln(x)\right) - 1 \right] = x \cdot \left(3 - \ln(x)\right)
$$

To maximize the profit, we find the derivative of $$ P(x) $$ with respect to $$ x $$ and set it equal to 0.

First, differentiate $$ P(x) $$:

$$
P(x) = x \left(3 - \ln(x)\right)
$$

Applying the product rule:

$$
P'(x) = \frac{d}{dx}\left[x\right]\cdot\left(3-\ln(x)\right) + x\cdot\frac{d}{dx}\left(3-\ln(x)\right)
$$

Since

$$
\frac{d}{dx}(x)=1 \quad \text{and} \quad \frac{d}{dx}(3-\ln(x)) = -\frac{1}{x},
$$

we get

$$
P'(x) = \left(3-\ln(x)\right) + x\left(-\frac{1}{x}\right) = 3 - \ln(x) - 1 = 2 - \ln(x)
$$

Setting the derivative equal to 0:

$$
2 - \ln(x) = 0 \quad \Longrightarrow \quad \ln(x) = 2
$$

This gives

$$
x = e^2
$$

Now, substitute $$ x = e^2 $$ into the demand function to determine the optimal price:

$$
p = 4 - \ln(e^2) = 4 - 2 = 2
$$

Thus, the hot dogs should be priced at

$$
\boxed{2}
$$

dollars each to maximize profit.
To maximize profit, the hot dogs should be priced at $2 each.

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