#1 ENG106
Homework Solution
Winter 2012
Question 1. Youve decided to buy a Toyota Prius because of the (recently) high price of gasoline. Like
every car, a Prius wears as it ages, providing less in psychological benefits and costing more for
maintenance and repair in future years. Therefore youd like to determine the optimal length of time to
keep the Prius before buying a new one.
a) What might be one or more possible objectives (wants)
Minimize average yearly cost
Maximize years of use
Maximize resale value
Minimize initial and annual CO2 emissions
b) List three or more possible constraints (needs)
Needs warranty throughout ownership period
Needs to be sold prior to major depreciation period
Availability during ownership period
Repair costs during ownership period
c) What are the decision variables for this problem?
How long to keep before buying a new Prius. (years)
d) List at least five pieces of information you might need in order to make a rational decision.
The yearly price costs under owning or leasing schemes.
The prices (and availability) of hybrid repair shops in the area
The trade in or selling value of the car
The average maintenance costs, with respect to year.
The average repair costs for hybrids, with respect to year.
e) Track down a reputable source for one of the five bits, and report the value. List complete
reference/contact information for the source.
The total price (including taxes and shipping) for a Toyota Prius in year 2000 reference dollars is
$19,746. Source: TE Lipman, MA Delucchi. Hybrid-electric Vehicle Design: Retail and Life Cycle Cost
Analysis. Analysis and Report Prepared for The Energy Foundation. April, 2003.
2.2 You deposit $2000 in a savings account that earns 8% simple interest per year. How many years will it
take to double your balance?
F = P (1 + N),
where F = $4000, P = $2000, and = .08
Solving for N = 12.5 years. (13 years)
(2.2)
If, you instead, deposit the $2,000 in another savings account that earns 7% interest compounded yearly,
how many years will it take to double your balance?
F = P (1 + )N
Where F = $4000. P = $2000, and = .07
Solving for N = 10.24 years. (11 years)
(2.3)
2.5 You are about to borrow $5,000 from a bank at an interest rate of 9% compounded annually. You are
required to make five equal annual repayments in the amount of $1,285.46 per year, with the first
repayment occurring at the end of year one. For each year, show the interest payment and principal
payment.
Interest Payment = Last Years Balance x Interest
Principal Payment = Annuity Interest Payment
Loan Balance = Last Years Balance Principal Payment
Period
0
1
2
3
4
5
Principal
Payment
0
$835.46
$910.65
$992.61
$1,081.94
$1,179.32
Interest
Payment
0
$450.00
$374.81
$292.85
$203.52
$106.14
Loan
Balance
$5,000
$4,165
$3,254
$2,261
$1,179
$0
2.8 Which of the following alternatives would you choose, assuming the interest rate of 8% compounded
annually?
Alternative 1: Receive $100 today;
Alternative 2: Receive $120 two years from now.
Alternative 1: P = $100
Alternative 2: P = $120 (P/F, 8%, 2) = 120 (0.8573) = $102.88
Alternative 2 is preferred
2.10a State the present worth of the following future payments:
P=?
= 7% , N = 6
$4,500
F = $4,500, 7% compounded annual interest, in 6 years time
F = P (1 + )N
, Solve for P
Or F*(P/F, ,n) = P, with = .07 and N = 6
F*(P/F, 7%, 6) = 4,500 * .6663 = $2,998
(2.3)
(using appendix B)
2.17 If $1000 is invested now, $1,500 two years from now, and $2,000 four
years from now at an interest rate of 8% compounded annually, what will be
the total amount in 10 years.
$1000,
0 years
F = ?,
10 years
$1500,
2 years
$2000,
4 years
P(1+ ) N = P(F/P, , N)
1000 (F/P, 8%, 10) + 1500(F/P, 8%, 8) + 2000 (F/P, 8%, 6) = F
1000(2.1589) + 1500(1.8509) + 2000 (1.5869) =
2158.9 + 2776.35 + 3173.8 = $8,109.1