In preparing for the upcoming holiday season, Fresh Toy Company $($FTC$)$ designed a new doll called The Dougie that teaches children how to dance. The fixed cost to produce the doll is $$100,000.$ The variable cost, which includes material, labor, and shipping costs, is $$29$ per doll. During the holiday selling season, FTC will sell the dolls for $$37$ each. If FTC overproduces the dolls, the excess dolls will be sold in January through a distributor who has agreed to pay FTC $$10$ per doll. Demand for new toys during the holiday selling season is extremely uncertain. Forecasts are for expected sales of $60,000$ dolls with a standard deviation of $15,000.$ The normal probability distribution is assumed to be a good description of the demand. FTC has tentatively decided to produce $60,000$ units $($the same as average demand$),$ but it wants to conduct an analysis regarding this production quantity before finalizing the decision.

$($a$)$

Determine the equation for computing FTC$\text{'}$s profit for given values of the relevant parameters $($e$.$g$.,$ demand, production quantity, etc.$).$

Using this equation, compute FTC$\text{'}$s profit $($in dollars$)$ when realized demand is equal to $60,000($the average demand$).$

$

$380000$

Correct: Your answer is correct.

$($b$)$

Modeling demand as a normal random variable with a mean of $60,000$ and a standard deviation of $15,000,$ simulate the sales of the Dougie doll using a production quantity of $60,000$ units.

What is the estimate of the average profit $($in dollars$)$ associated with the production quantity of $60,000$ dolls? $($Use at least $1,000$ trials. Round your answer to the nearest integer.$)$