Answer each question in the space provided. Each of the
nine questions will count nearly the same.
Formulas for Questions 1-3: Exponential growth ,
Exp onential decay ,
1. Skeletons were found at a construction site in San
Francisco in 1989. The skeletons had lost 12% of the amount
of carbon-14 that would be found in a living person. In 1989, how old were the
skeletons? (The decay constant
k for carbon-14 is 0.00012)
Losing 12% of carbon-14 means 88% of original amount was
still present
2. A person invests $2000 in an account that compounds
interest continuously at 6% per year. How long will it
take for the account to double the amount originally invested? Give your answer
to the nearest tenth of a year.
3. A colony of mice has a current population of 120 and
grows continuously at the rate of 20% per month.
(i) Write the equation that gives the number of mice in this colony as a
function of time, t, in months.
(ii) De termine how many mice there will be two years ,
i.e., 24 months, later?
4. Chumps Refreshment Stand sold a three-item lunch
consisting of a hamburger, a bag of fries, and a soda for
$5.00. But then Chumps had to increase the price for fries and sodas. The price
of fries doubled, and the price
of sodas increased 50%, so that the three-item lunch now costs $7.00. Also,
before the price increase, fries cost
50% more than soda. What are the current prices of each of the hamburger, fries,
and soda at Chumps?
5. A vacation riverboat travels 46 km downstream in two
hours. It travels 51 km upstream in 3 hours. Find the
speed of the boat in still water and the speed of the river.
6. (a) Write the system of equations as an augmented
matrix.
x + 2y + z = 2
- 2x - 2y + 4z = -14
3x + 2y - z = 10
(b) Convert the augmented matrix to row-echelon form. As a
reminder, the row-echelon form of the augmented
matrix looks like 
You will provide values
for a , b, c, e, f, and g.
(c) Using your result for (b), solve for (x, y, z).
Work for all of (a) – (c) to be done below.
7. The circle equation in standard form is (x – h)2 + (y
– k)2 = r2.
(a) For the circle described by (x + 2)2 + (y + 4)2 = 7
(i) Give the coordinates of the circle ’s center:
(ii) State the exact value of the radius of the circle:
(b) For the circle described by x2 + y2 + 8x - 6y + 9 =
0
Write the equation of
the circle in standard form.
Give the coordinates of the circle’s center:
State the exact value of its radius:
Carefully sketch the circle on the grid at
the right.
8. Graph the fol lowing system of inequalities. Shade the
region which satisfies all four inequalities and give the
coordinates of the four vertices.
x + y < 9
x + 3y ≥ 6
x ≥ 3
y ≥ 0
Vertices: (3, 1), (3, 6), (9, 0), (6, 0)
9. For Question 9 answer one of the following two
problems. If you attempt both of them, please circle the letter,
(a) or (b), of the one you want to be graded.
(a) A computer manufacturer needs to make up to a total of
100 laptop and tabletop computers each week. It
takes 3 hours to make a laptop and 2 hours to make a table top computer, and
there are 240 hours of labor
available for making the computers. The weekly profit is $80 per laptop and $60
per table top. (a) How
many laptop, and how many tabletop computers should they make each week to
maximize their profit? (b)
What is the weekly maximum profit this manufacturer can make? Use the grid
provided to demonstrate
how you arrive at your conclusion. Be sure to show some steps taken in solving
this problem by writing out
the objective function and constraints in the places indicated.
Objective Function:
x: number of laptop computers
y: number of tabletop computers
P = 80x + 60y
Constraints:
x + y ≤ 100
3x + 2y ≤ 240
x ≥ 0; y ≥ 0
Vertex Profit
(0, 0) P = 80(0) + 60(0) = 0
(0, 100) P = 80(0) + 60(100) = $6000
(40, 60) P = 80(40) + 60(60) = $6800*
(80, 0) P = 80(80) + 60(0) = $6400
*Maximum profit of $6800 when 40 laptop and 80 tabletop
computers are made
(b) Marita will take a test that has questions in group A
worth 9 points each and questions in group B worth 12
points each. A total of at least eight questions must be answered. From her
experience on practice tests,
Marita knows that group A questions take 8 minutes each and group B questions
take 16 minutes each, and
that she is allowed a maximum of 80 minutes for the test. (a) How many questions
from each group must
she answer correctly to maximize her score? (b) What is her maximum possible
score? Use the grid
provided to demonstrate how you arrive at your conclusion. Be sure to show some
steps taken in solving
this problem by writing out the objective function and constraints in the places
indicated.
Objective Function:
x: number of question from group A
y: number of questions from group B
S = 9x + 12y
Constraints:
x + y ≥ 8
8x + 16y ≤ 80
x 
≥ 0; y ≥
0
Vertex Profit
(6, 2) S = 9(6) + 12(2) = 78
(8, 0) S = 9(8) + 12(0) = 72
(10, 0) S = 9(10) + 12(0) = 90*
*Maximum score of 90 points when 10 group A and 0 group B
questions are answered.
