Pre-Calculus 12 – Group Work
Polynomials
For the following group assignment fully worked out answers are necessary for full
value. All work must be shown and solutions must not rely entirely on graphing
technology. Clearly state the domain for each question.
1. In a coal mine, there are x men in each shift at the coalface, and the output of
2 2 1 3
coal is given by T x x tones of coal.
3 30
a) What are the practical restrictions on the values of x ?
b) What is the optimum number of men per shift?
2. The owner of a small business estimates that the profit form producing x items
is given by the function P( x) 0.003 x 3 1.5 x 2 200 x 1000 . This function is
based on current production levels, which cannot exceed 250 items due to
limited space and resources. How many items should be produced to maximize
the profit?
3. For the years from 1995 to 2001, the number of women (in thousands) enrolled
in graduate school in the United States can be modeled by the function
W (t ) 2t 3 14 t 2 50 t 696 ,where t is the number of years since 1995.
Determine the year when the number of female graduate students reached 800
thousand.
4. For a certain windmill, the power produced when the wind speed is s m/s can be
modeled by the function: P( s ) 0.2s 3 5.7 s 2 38 .8s 282 , where P(s) is in
kilowatts (kW) and s (4,16) .
a) At what wind speed does the power produced by the windmill reach 300kW?
b) At what wind speed is the power production a maximum?
5. For the period of 1890 – 1990, the American Indian, Eskimo, and Aleut population
P (in thousands) can be modeled by: P(t ) 0.005 t 3 0.33t 2 11 .3t 1245 .8 , where
t is the number of years since 1890.
a) In what year did the population reach 500 thousand?
b) When did this population reach a minimum value?
6. In the last year (Starting January 1st), the volume of water (in megalitres) in a
particular reservoir after t months could be described by the model
V (t ) t 3 30 t 2 131t 250 . The reservoir authority rules that if the volume
falls below 100ML , irrigation is prohibited. During which months, if any, was
irrigation prohibited in the last twelve months?
Pre-Calculus 12 – Group Work
Polynomials
Interesting Fact: (bonus if you have time to try it…)
Bhaskara was an Indian mathematician who lived during the twelfth century AD. In his
book Siddhanta Siromani (The Gem of Mathematics), he solved the cubic equation
x3-6x2=-12x+35
using these steps:
x3 – 6x2 + 12x – 8 = 27
(Add (12x-8) to both sides so that each side becomes a perfect cube)
(x-2)3= 33
x-2 = 3
x=5
a) Use Bhaskara’s method to solve the equation x3 + 3x2 = -3x + 7.
b) Can Bhaskara’s method be used to solve any cubic equation? If not, what types of
equations are solvable with his method?