A geologist you spoke with is concerned about the rate of land erosion around the base of a dam.

A geologist you spoke with is concerned about the rate of land erosion around  the base of a dam. Another geologist is studying the magma activity within the  earth in an area of New Zealand known for its volcanic activity. One of the shortcuts they apply when doing calculations in the field is to use synthetic division. After they explain the concept and process to you, you decide to try and solve some problems.

1. Use synthetic division to determine if the first set of numbers are zeros of the given polynomial
a. -3, 2. f(x) = 3x^3  +  5x^2 - 6x + 18.
a. -4, 2. f(x) = 3x^3 +11x^2 - 2x  + 8.

2. Given the polynomial f(x) = 2x^3  -5x^2-4x +3, find the solutions if the function is completed as  
          a) f(x)= 0
          b) f(x +2)=0
          d) f( 2x) =0

3. Today you visited the R&D (Research and development) development of a  pharmaceutical manufacturer. The people in this lab have to calculate the recommended dosage of the products they are  developing. Medicine  is  expressed   as  a  function  of  the   concentration   in  parts   per  million, with  the  potency   of  the  dosage   declining   over  a period   of  time. The  function  N(t) = ( 0.8t + 1000) /  (5t +4 ), where t= > 15 gives   the   body   concentration N(t), in  parts  per million , of  a  certain  dosage  of  medication  after  time t, in  hours.  Find  the  horinzontal   asymptote   of  the  graph   and  complete   the  following   statement: N(t) approaches ? as  t approaches   infinity. Explain  the meaning   of   the  answer  to  a)  in  terms   of   the   medical  dosage    application.

4. A retailer  you   spoke    with  in New York  City's  fashion  district   imports  haute   couture  from  European  designers . One  of  the   accommodations  which  must   be considered   when  importing   fashion  from   other  countries  is  the  difference  in  the  size   charts. A function  that  will   convert   dress   sizes  in  the United  States  to those   in  Italy  is  g(x) =  2(x + 12). Find  the  dress  sizes   in  Italy   that  correspond  to  sizes  6, 10, and  14  in  the  United  States. Find   a formula   for  the  inverse  of  the  function . Use   the  inverse  function  to  the  dress  sizes  in  the  United  States   that   correspond   to  36, 44 , and  60  in Italy.

5. Last  week's   profits   from  a  dry  cleaners  was  $2000.  suppose  the   $2000 is  invested    at   interest   rate  k,   compounded  continuously,  and  grows   to  2983.65  in  5 years.

a. what  is  the  interest  rate?, b. find   the exponential  growth   function, c.  what  will  the  balance   be  after  10 years ?,  c. when  will  the  $ 2,000  double?