Project #5077 - pre cal math work

1. (15 points)

(a)  Section 5.4, number 6.

(b) Section 5.4, number 10.

 

2. (5 points)  Find all solutions to the equation, .  Check your solution(s).

 

3.  (10 points)

(a) Perform the indicated operation and express the result in the form a + bi.

(1 - 2)(2  + 3)

 (b) Find all real numbers x and y that satisfy the equation:  -2x - 4 + 2yi  =  3x – 2i

 

4.  (10 points) section 6.2, number 6.

 

5.  (5 points)  Section 6.3, number 6.

 

6. (10 points)  Section 7.1, number 4.

 

 

7. (10 points)  Suppose you deposited $2000 in a savings account with an annual rate of interest of 1% compounded continuously.  How much money will be in the account in 10 years?

 

8. (10 points)

(a)  Section 7.4, number 22.

(b)  Solve for x:    .

(c)  Determine the exact value of   ln e5. (Do not give a calculator estimate.)

 

   Before you do number 9 study the following example 

Example.  (Example 2 of section 7.3 revisited)

       A person deposits $1,000 in a bank account which pays 8% annual interest compounded continuously.  How many years will it take for the amount of money in the account to double.

 

The mathematical model, using continuous compound interest, of this problem is

 

                                 A = Pe.08n

 

In this case, P = $1,000 and we want to find n when A = $2,000.

 

       2000 = 1000 e.08n                                   divide both side by 1000

       2 = e.08n                                                  take the natural log (ln) of both sides.

       ln 2 = ln(e.08n)                                         ln 2 = .08n ln e                                  Simplify using ln e = 1

                                                                        (Do you see why we took the ln of both sides as                                                              opposed to log10 of both sides)

       ln 2 = .08n                                                divide both sides by .08

  so that n =

 

      

So it takes about 8.6 years (or 8 years and 8 months) for the money in the account to double.

 

 

A side remark.  A rough estimate of the number of years it takes for money to double is to divide the percent into 72.  Here 72/8  = 9 (years), which is close to the above solution  8.6 of years.  This estimation process is called the law of 72.

 

9.  (10 points)  A person deposits $2,000 in a bank account which pays 1% annual interest compounded continuously.  How many years will it take for the amount of money in the account to double?  Use the above process to determine an exact solution and the check your solution using the estimate of the “law of 72”.

 

 

10.  (15 points) A fax machine is purchased for $5,800.  Its value each year is about 80% of the value of the preceding year.  So after t years the value, in dollars, of the fax machine, V(t), is given by the exponential function

      V(t) = 5800 (0.8)t.

a)      Give a sketch of the graph of the function V(t).  Your graph can be a “rough draft”.

b)      Determine the value of the fax machine in years 0, 1 and 4. to the nearest tenth.

c)      Assume that the company decides to replace the machine when the machines values reduces to $500. In how many years will the machine be replaced.

 

  Bonus questions

 

1.  Section 7.2 number 4

 

2.  A trainee is hired by a computer manufacturing company to learn to test a particular part of a personal computer after it comes off the assembly line.  The learning curve for an average trainee is given by  .

a)      How many computers can a trainee be expected to test after 2 days of training?    Round answers to the nearest integer.

b)      How many days will it take until an average trainee can be expected to test 40 computers per day?  Round answers to the nearest integer.

c)      Does N approach a limiting value as t increases without bound?  In other words as time, t, increases what is the upper limit for the number of computers that an average trainee can be expected to test?  Explain.

 

   3.  Many countries have a population growth rate of about 3% or more a year.  Assume        country X has a population growth rate of 3%.   How many years will it take the population  of country X to double?  Use continuous compound growth.

  

 

 

 

 

 

Subject Mathematics
Due By (Pacific Time) 04/28/2013 06:00 pm
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