Take Home Test $11.$ Radiant requires cargo planes for the transportation of their goods and has approached VIP Transportation Company for this specific purpose. VIP has presented two options of cargo planes $-$ WT$88$ and BH$54,$ and has provided comprehensive cost$-$related information for both of them. Personnel CostFuel CostMaintenance CostWT$88150012501150$BH$54155015501100$ Radiant Airlines has a clear plan in place for allocating funds towards personnel, fuel, and maintenance expenses. They are $-$ Radiant has $$250,000$ to spend on personnel costs, $$160,000$ on fuel, and $$80,000$ in maintenance costs. To ensure maximum efficiency, they require a minimum of three planes for each type, with two options available: the WT$88,$ boasting a weight capacity of $55$ tons, and the BH$54,$ capable of transporting up to $85$ tons. Through the implementation of Linear Programming, help Radiant Airlines determine the necessary information required for optimal operations.$($a$)$ What is the maximum amount Radiant can carry, given their budget constraint?$($b$)$ How many WT$88$ and BH$54$ radiant should hire to carry the maximum amount?Keep in mind that the number of WT$88$ should be at least twice the number of BH$54.2.$ Radiant requires cargo planes for the transportation of three distinct goods: A$,$ B$,$ and C$.$ They have contacted VIP Transportation Company to arrange the rental. It has been communicated to Radiant that VIP has two types of cargo planes available, with $55$ of each type. Furthermore, VIP has shared weight$-$related information for the two types of planes, WT$88$ and BH$54.$ Goods AGoods BGoods CWT$88$ Can carry$18\mathrm{.}5$ tons$12$ tons$20$ tonsBH$54$can carry$15$ tons$20$ tons$15\mathrm{.}5$ tons To transport their cargo, Radiant needs to move $300$ tons of Goods A$,300$ tons of Goods B$,$ and $350$ tons of Goods C$.$ In order to do so$,$ they will require at least two WT$88$ and two BH$54$ planes. The cost for each WT$88$ plane with VIP is $$1600$ and each BH$54$ is $$1250.$Under the above circumstances, find out the following using Linear Programming: $($a$)$ What will be the minimum cost to transfer the required goods?$($b$)$ How many of WT$88$ and BH$54$ Radiant should hire? $3)$ Paul has to buy two types of Yogurts$-$ Brand A and Brand B for children. He knows the following regarding the Vitamins the yogurts have: Vitamin AVitamin BVitamin CBrand A$126\mathrm{.}58\mathrm{.}5$Brand B$125\mathrm{.}510\mathrm{.}5$ It is imperative that Paul ensures the children receive a minimum of $200$ units of Vitamin A$,150$ units of Vitamin B$,$ and $150$ units of Vitamin C$.$ This requirement must be met even if it means purchasing more expensive Brand B yogurts, which cost $$2\mathrm{.}25$ per unit compared to the $$1\mathrm{.}20$ per unit of Brand A$.$ Furthermore, there must be twice as many Brand B yogurts as Brand A yogurts, and currently, only $5$ Brand A yogurts are available at the store.Using linear programming find the minimum cost of buying the yogurts to provide the required units of vitamins. How many of each type of yogurt Paul should buy to keep the cost at the minimum. $4.$ An electric vehicle manufacturing company builds two types of cars, an X$8$ and an A$3.$ Car manufacturing has three stages: $1,2,$ and $3.$ X$8$ takes an hour in Stage $1,$ an hour in Stage $2,$ and $30$ minutes in Stage $3.$ On the other hand, A$3$ takes an hour and a half in Stage $1,45$ minutes in Stage $2,$ and another half an hour in Stage $3.$ The company must manufacture at least one car of each type per day to meet the market demand. However, each day, they have only seventeen and a half hours allocated for Stage $1,$ fourteen hours for Stage $2,$ and sixteen hours for Stage $3.$ If the profit for each X$8$ is $$1800$ and each A$3$ is $$3000,$ then find out how many of each type of car the manufacturer should produce daily to maximize the profit. Keep in mind that due to market limitations$,$ they only want to manufacture up to five of each type of car.