College Algebra
oscrh9
algebracommon2015.docxMATH133 NAME:___Oscar Hernandez________
Individual Project Assignment
(Show your work for these calculations. Review http://www.purplemath.com/modules/mathtext.htm to see how to type mathematics using the keyboard symbols.)
Problem 1 – Photic Zone
a. Choose a value of k between 0.025 and 0.095.
Problem 2 – Compound Interest
For discrete periods of time (like once per year, twice per year, four times per year, twelve times per year, 365 times per year, etc.), the English terms we use to describe these, respectively, are annually, semiannually, quarterly, monthly, daily, etc. The formula for calculating the future amount when interest is compounded at discrete periods of time is:
Page 1 of 5
A is the amount you will have after t years the money is invested, P is the principal (the initial amount of money invested), r is the decimal equivalent of the annual interest rate (divide the interest rate by 100), and n is the number of times the interest is compounded in ONE year. For the compounding continuously situation, the formula is:
A is the amount you will have after t years for principal, P, invested at r decimal equivalent annual interest rate compounded continuously.
Based on the first letter of your last name, choose values from the table below for P dollars and r percent.
|
If your last name begins with the letter |
Choose an investment amount, P, dollars between |
Choose an interest rate, r, percentage between |
|
A–E |
$5,000–$5,700 |
9% - 9.99% |
|
F–I |
$5,800–$6,400 |
8% - 8.99% |
|
J–L |
$6,500–$7,100 |
7% - 7.99% |
|
M–O |
$7,200–$7,800 |
6% - 6.99% |
|
P–R |
$7,800–$8,500 |
5% - 5.99% |
|
S–T |
$8,600–$9,200 |
4% - 4.99% |
|
U–Z |
$9,300–$10,000 |
3% - 3.99% |
Suppose you invest P dollars at r% annual interest rate. (Correctly round your answers to the nearest whole penny (two decimal places) and show the intermediate steps in all these calculations for full credit.)
a. How much will you have in 8 years if the interest is compounded quarterly?
b. How much will you have in 15 years if the interest is compounded daily?
c. How much will you have in 12 years if the interest is compounded continuously? Use e ≅ 2.718282.
Problem 3 – Newton’s Law of Cooling
According to Sir Isaac Newton’s Law of Cooling, the rate at which an object cools is given by the equation:
b. What do you think k in this formula represents?
c. Freezing is 32o F. How many hours will it take for this dessert to freeze? (Correctly round your answer to two decimal places and show the intermediate steps in your work.)
Problem 4 – Medicare Expenditures
The following Medicare data which represents the Medicare expenditures for years after 2000 in the United States is from the U. S. Census Bureau.
|
Actual Year |
Years after 2000 ( x ) |
Medicare Expenditures (in Billions of Dollars) |
|
2004 |
4 |
311.3 |
|
2006 |
6 |
403.1 |
|
2007 |
7 |
431.4 |
|
2008 |
8 |
465.7 |
|
2009 |
9 |
502.3 |
A natural logarithmic regression function representing this data model is of the form:
This data can be closely modeled by the logarithmic regression function:
a. Choose a value for x between 15 and 30 (it does not have to be a whole number). Based on this natural logarithmic function, what will be the expenditure for Medicare in the year represented by your chosen value of x? (Correctly round your answer to one decimal place, this is tenths of billions of dollars. Also show the intermediate steps in your work.)
b. Based on this formula, in how many years after 2000 will the Medicare expenditures be $700 billion? (Correctly round your answer to one decimal place and show the intermediate steps in your work.)
c. Using Excel or another graphing utility and the values from the table above, draw the graph of this function:
On your graph does this data seem to represent a natural logarithmic function? Explain your answer. Is there another function type that we have studied that seems to more closely match the data? Explain your answer.
d. In an English sentence, state the types of transformations of the natural logarithmic function:
that will result in the function:
Problem 5 – Richter Scale
a.
|
E |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
I,
-kx
0
e
I
=
I
0
I
I
1000
I
0
=
01
.
0
I
I
0
=
-0.062x
e
0.01
=
nt
÷
ø
ö
ç
è
æ
+
=
n
r
1
P
A
rt
e
P
A
=
kt
m
m
e
T
T
T
-
-
+
=
)
(
T
0
T
0
T
m
T
m
T
)
ln(
)
(
g
x
b
a
x
+
=
)
ln(
9582
.
229
5904
.
9
)
(
E
x
x
+
-
=
)
ln(
)
(
f
x
x
=
4
.
4
0
10
=
E
÷
÷
ø
ö
ç
ç
è
æ
@
0
log
6667
.
0
E
E
M
;
)
log(
)
10
log(
b
a
a
b
+
=
´
,
3
.
6
29897
.
6
6
.
5
69897
.
0
6
.
5
)
5
log(
)
10
5
log(
6
.
5
@
@
+
@
+
=
´
0
E
E
x
=
(
)
x
x
M
log
6667
.
0
)
(
@
6
10
5
.
0
´
6
.
1
10
5
.
0
´
8
10
0
.
1
´
6
.
3
10
0
.
1
´
10
10
5
.
1
´
6
.
5
10
5
.
1
´
12
10
5
.
2
´
6
.
7
10
5
.
2
´
14
10
99
.
1
´
,
6
.
9
10
99
.
1
´
4
.
4
10
