Math 55(31) - Warren D. Smith - Homework #4 - due Tues 18 Feb 2003
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Exercises:

This time I'm going to give you exercises like those in the book, but
not the same (I'm making them up myself this time).

Ex 1.  (Like ex 1 p508 9A but not same.)
Draw axes and plot and label these points:
(0,5),  (-2,7),  (3,2),  (4,-3),  (-3, 5).

                      ^y
                      |
                      9
                      8
                    *(-2,7)
                      6
            (-3,-5)*  *(0,5)
                      4
                      3
                      2  *(3,2)
                      1
----------------------+123456789---------->x
                      |
                      |
                      |   *(4,-3)
                      |
                      |
                      |
                      |
                      |

Ex 2.  Consider the function  F(x) = (x+7)/(38-x).
(a) What is the NAME of this function? 
              ANS F.
(b) What letter in "F(x)" is playing the role of the INPUT
to this function?
              ANS x.
(c) What is F(23)?
              ANS (23+7)/(38-23) = 30/15 = 2.

                                      (x+3)/2  if x>5
Ex 3.  Consider the function  G(x) = {
                                      -77      if x<5
(a) What is the NAME of this function? 
             ANS G.
(b) What is G(7)?
          ANS   since 7>5 we use the first case getting (7+3)/2=5.
(c) What is G(2)?
          ANS   since 2<5 we use the 2nd case getting -77.

Ex 4.   (Like ex 23 p522 9B but not same.)
The price of a bulldozer is $50000 today.
But 5 years ago, it cost $41000.  
(a) Assuming the price depends approximately LINEARly on time,
how much will a bulldozer cost in 4 years?
(b) If t is "number of years starting now"
(for example, 2 years ago t = -2) then what is a FORMULA
for  PriceOfBulldozer(t)?   Hint:  your formula should have the
property that    PriceOfBulldozer(4) = Your answer to (a).
   ANS(b).   PriceOfBulldozer(t) = 50000 + 1800 t.
sanity checks: PriceOfBulldozer(0) = 50000; PriceOfBulldozer(-5) = 41000.
How got it: slope=S=(50000-41000)/5=rise/run=1800.
Line formula: S t + K  is  1800t + K.
We choose K so formula works at one of our datapoints (0, 50000) or (-5, 41000).
   ANS(a).    PriceOfBulldozer(4) = 50000+1800*4 = 57200.

Ex 5.  (Like ex 19-22 p522 9B but not same.)
Consider the equation     y = 2 x - 3.
(a) Sketch an x-y plot of this function. 
(b) What is its SLOPE?                ANS: 2.  
(c) What is its y-intercept?          ANS -3.

Ex 6.   (Like ex 17 p522 9B but not same.)
Suppose it costs $34000 to buy a condo, and then it costs
$5000 per year in "fees" for every year thereafter.
Let t be the total number of years since you buy the condo.
What is a FORMULA  for  TotalMoneyYouPaidSoFar(t)?
Hint: Your formula should have the
property that     TotalMoneyYouPaidSoFar(0) = 34000.
  ANS  TotalMoneyYouPaidSoFar(t) = 34000 + 5000 t.

Ex 7.  (Like ex 21 p468 8A, but more accurate).
The population of Earth is 6.5 billion.
It is doubling about every 40 years.
The total surface area of the Earth (only counting habitable land)
is about                                    14
          100 million square kilometers = 10  square meters.
Assume to grow enough food and produce enough resources 
for an average human to live off of, requires a 62meter X 62meter
square patch of land (about half the size of a football field).
HOW MANY years have we got (if this doubling rate continues) 
before humankind can no longer live?  

ANS:
    Only counting habitable land, e.g., not antartica and
    not lakes (Counting ALL land would give 148 million square km.
    Land is about 20% snow/ice, 20% mountains, 10% land with no topsoil...)
    We have total land per person right now
       = 10^14 / (6.5 * 10^9) = 15384.6 square meters.

    But supposing it takes a 62*62 meter plot (62*62=3844 square
    meters) to support an average person, that means the human
    population cannot grow by a factor of more than
           15384.6 / 3844 =  4.002237253
    before it is in deep doodoo. 

   Note 4.0022 is about 2*2, meaning we can double just TWICE before
   the doodoo point.  If each doubling takes 40 years, that
   means we have 80 years left.

Obviously it is hard to be sure these input numbers are exactly right,
but you can see humankind is going to start getting into serious
problems in this century if we don't change our habits pretty damn
quick, and the fact things don't look too bad right now, is irrelevant.