Question

5.  Find the two roots of each of the following functions (that is find the two X values that make Y = 0).  This skill may be useful in assignment 11.  EXAMPLE:  Y = 3X² -11X +6 is the product of (3X -2)(X -3).  If you let Y = 3X-2 then X = 2/3 will make Y = 0.  If you let Y = X-3 then X = 3 will make Y = 0.  Thus both X = 2/3 and X = 3 are roots.   Show the algebra involved.

a. Y = -4X² -14X -6.  The two roots are X = ______ and X = ______.

6.  Exponential functions are useful in business and economics.  Lesson 7 discusses them.  Show how the values are entered into your functions and also calculate the amounts of each of the following:

            a1. You learn on the business channel that inflation was about 0.75% last month.  Assume this rate is maintained each month for a year.  What will the annualized rate be?  EXAMPLE:  A rate of 0.1% per month represents (1 + 0.001)¹²-1 = 0.0121 or 1.21% annually.

            a2.  Population increased 8 1/2% last year.  What average monthly growth rate is implied?

EXAMPLE: A 2% growth rate for the year would require 1.02 = (1 + r)¹².  Solve this for r:  ; r = .00165 or .165% per month on average.

            b1.   F = Pe^rt, which assumes continuous compounding, says that the Future value (F) of an amount (P) invested today at an annual rate (r), expressed as a decimal for the time (t) in years, is given by the function.  EXAMPLE: invest $100 at the annual rate of 5 1/2% for 6 years and 3 months and you should get back (at the end of the time), F = $100e^{(0.055)(6.25)} = $100e^(0.3438) = $100(1.4102) = $141.02.   Some auto dealerships are advertising, “72 months, no interest, no payments” However, on the maturity date you have to pay the whole amount owed.   If you are even a minute late, then you owe all the interest you avoided, too.  You want a $50000 auto for which, after down payment,  you qualify for a $40000 loan as advertised.   Six percent per annum will be assessed if you are late paying.  What total amount will you owe if you are late?   

  

            b2.  Alternatively, if a borrower tells you that he needs a loan for 6 years and 3 months and will pay you an annual rate of 5 1/2% for the loan, but will only give you $141.02 back at the end of the loan term, you should only loan him $100 today.  Here is a loan proposition more in line with current rates.  A borrower agrees to pay you 4.5% annually for 3 years and 3 months.  At the end of the term he will make a balloon payment of $9000 to repay the loan and interest.  What amount (P) does the formula P = F/e^rt indicate you should loan this prospect?

Answer 1

5 a y = - 42- 14 X - 6 . 4x2+12x+2x+6) [1x Cx+3)+2 x+ 3) +2) (2+3) .Y=Owhen ther 4x+2) o ar, K+3) = 0 then t 3= 0 then :-3 Hence the two rwots ace x-2 and X= -3