Please answer all questions and provide detailed step by step solution for each question and show step by step calculation

Functions 1. Suppose that we are establishing a perpetuity that pays $6,000 at the end of every six months. (a) Suppose the endowment can be invested at 7.2% p.a. compounded semi-annually. How much money is needed to establish this per- petuity? (With the first payment at the end of the first six months.) Give your answer rounded to the nearest dollar. (b) Suppose we can only afford $150,000 to establish this perpetuity. If we defer the first payment for long enough we will be able to meet the objective of $6,000 payment at the end of every six months. How many six-month periods will we need to defer this perpetuity? (c) Suppose that we are not happy with the long wait (deferral pe- riod) and would like to seek better investment options so that the first payment of $6,000 can be made at the end of the first six months, with $150,000 as the initial investment. What is the per annum nominal interest rate, compounded semi-annually, that would be needed? 2. Anne negotiates with a bank to repay a loan of $280,000 in a series of level payments at the end of every quarter over 20 years with the interest at the rate of 5.6% p.a. compounded monthly. How much in total will she pay each quarter? Give your answer rounded to the nearest dollar. Algebra (The following question is by no means related to your lecturer nor giving you any recipe for pancakes/cakes.) 3. Ameilia is going to run a company that makes pancake mix and cake mix. Each kilogram of pancake mix uses 300g of flour and 50g of short- ening. Each kilogram of cake mix uses 180g of flour, 50g of shortening and 180g of sugar. Suppliers can deliver at most 2400kg of flour, at least 200kg of shortening, and at most 540kg of sugar. If the profit per kilogram is $0.4 for pancake mix and $0.2 for cake mix, how many kilograms of each mix should be made so that she can earn maximum profit? Formulate this problem as a linear program in standard form, but do not solve it. Clearly define the variables you use, the objective function and all of the constraints