Problem $1$: Non$-$linear Optimization $(15$ points$)$

You are the General Manager of Northern Virginia Motors that specializes in the sales of station wagons and light trucks. Your company has a strong reputation in the market but customer demand remains somewhat sensitive to price. Based on historical data analysis, your marketing, team has come up with the demand curves for trucks and for wagons as follows:

Truck Demand $($in unit$)=500-18$ x $($Truck Price is in thousand $$)$

Wagon Demand $($in unit$)=400-11$ x $($Wagon Price in thousand $$)$

Your purchasing prices from automakers are respectively $$20,000$ per truck and $$25,000$ per wagon.

Before you sell your vehicles in the market, you need to prepare them. It takes $3$ hours to prepare a truck and $2$ hours for a wagon. You currently have a staff with a total of $250$ hours labor hours that you can deploy for the prep work and it costs $$30$ per hour for each hour used. Also, you have a line of loans of $$2$ Million from the bank available for you to finance your vehicle purchases and the interest is $5\%$ for the amount used.

$(1)$ Base Case Evaluation. You are thinking about pricing each truck at $$26,000$ and each wagon at $$32,000.$ Calculate the following using the above information $(3$ points, $(1)/(2)$ points each for each number$)$

$-$ How many vehicles do you need to order to satisfy the demand?

# of trucks:

ll of wagons:

How big a loan that you need to purchase the vehicles lin $$000)?$

How many labor hours do you need in preparing the vehicles?

What's your total revenue $($in $5000)?$

I

$-$ What's your profit $(5000)?$

$($Do not forget about labor cost and interest charge$)$