At the beginning of the first day (day 1) after grape harvesting is completed, a grapegrower has 8000 kg of grapes in storage. At the end of day n, for n = 1, 2, . . . , the grape grower sells 250n/(n + 1) kg of their stored grapes at the local market at the price of $1.50 per kg.During each day the stored grapes dry out a little so that their weight decreases by 2%.Let wn be the weight (in kg) of the stored grapes at the beginning of day n for n ≥ 1.Find a recursive definition for wn. Draw a timeline. Find the value of wn for n = 1, 2, 3. Let rn be the total revenue (in dollars) earned from the stored grapes from the beginning of day 1 up to the beginning of day n for n ≥ 1.Write a program to compute wn and rn for n = 1, 2, . . . , num where num is entered by the user, and display the values in three columns: n, wn, rn with appropriate headings.Run the program for num = 20. (Use format bank.) Use  program to determine how many days it will take to sell all of the grapes.//This is all…

You wish to drive from point A to point B along a highway minimizing the time that you are stopped for gas. You are told beforehand the capacity C of you gas tank in liters, your rate F of fuel consumption in liters/kilometer, the rate r in liters/minute at which you can fill your tank at a gas station, and the locations A = x1, ··· , B = xn of the gas stations along the highway. So if you stop to fill your tank from 2 liters to 8 liters, you would have to stop for 6/r minutes. Consider the following two algorithms: (a) Stop at every gas station, and fill the tank with just enough gas to make it to the next gas station. (b) Stop if and only if you don’t have enough gas to make it to the next gas station, and if you stop,fill the tank up all the way. For each algorithm either prove or disprove that this algorithm correctly solves the problem. Your proof of correctness must use an exchange argument.

2. A whisky distiller named Taketsuru Masataka puts 8000L of whisky into barrels for aging, with 200L of whisky per barrel. Each year the volume of whisky in each barrel reduces by 2% due to evaporation (this is known as the angels' share). For the first 11 years after barreling, Taketsuru sells 2 barrels for 20 dollars per litre at the end of each year. Write bn for for the volume of whisky in each barrel at the beginning of year n, write Un for the total volume of whisky in storage at the beginning of year n, and rn for the total revenue from selling whisky at the beginning of year n. (a) Find a direct formula for bn. (b) Calculate the values of v₁, v2, V3 and then find a recursive definition for Un. (c) Find a recursive definition for rn. (d) Write a MATLAB program to compute Un and rn for n = 1, 2 ... 12 and display the values in three columns n, rn, Un with appropriate headings. (e) At the end of the 12th year Taketsuru finds that he can sell 12-year old whisky at the higher price…