1. Calculate the rent of a decreasing annuity at 8% interest compounded quarterly with payments made every quarter-year for 12 years and present value $200,000. (Show work)
Let P be quarterly payment. Theree will be total 12*4 = 48 installments.
As payment is quarterly, we will have i = 8/400
Hence, present value of 1st installement =
Present value of 2nd installment =
Present value of 3rd installment = and so on up to 48 installments.
Sum of these installments will be equal to 200000 USD.
Hence,
We see that right hand side is geometric series having first term as P/(1+i) and common ratio as 1/(1+i). We have to find out sum of 48 terms. Hence,
Putting i = 8/400 = 0.02, we get
Hence, quarterly payment = $6520.37
2. An airline company wants to fly 1400 members of a ski club to Colorado. The airline owns two types of planes. Type A can carry 50 passengers, requires 3 flight attendants, and costs $14,000 for the trip. Type B can carry 300 passengers, requires 4 flight attendants, and costs $90,000 for the trip. If the airline must use at least as many tpe A planes as type B and has available only 42 flight attendants, how many of each type should be used to minimize the cost for the trip?
Let us suppose that type A plane makes x trips and type B plane makes y trips.
Hence, cost = 14000x+90000y …. (1)
Cost is to be minimized.
Hence, number of passengers flown = 50x+300y
As 1400 passengers have to fly, we should have
50x+300y≥1400 …. (2)
Flight attendants used = 3x+4y
Number of available flight attendants = 42.
Hence, 3x+4y≤42 …. (3)
If the airline must use at least as many type A planes as type B, we should have x>y … (4)
We should have x≥0,y≥0.
Permissible region is given by triangle ABC.
Cost at A(6,6) = 14000*6+90000*6 = 624000.
Cost at B(4,4) = 14000*4+90000*4 = 416000.
Cost at C(10, 3) = 14000*10+90000*3 = 410000.
Hence, cost is minimum when type A plane makes 10 trips and plane B makes 3 trips.