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1)
Solve the following equations algebraically. You must show all your work.
a)
3
x 53
Answer: x = 512
Show your work here: 3x = 3 + 5 = 8  x = 8³ = 512
b)
3
x2
 27
Answer: x = 9
Show your work here: x = 272/3 = (3³)2/3 = 3² = 9
c)
3
3x  2
1
4
x 1
Answer: x = 8³ = 512
Show your work here: x = 8³ = 512
2)
Solve algebraically and check your potential solutions:
x2  x0
Answer: x = 8³ = 512
Show your work here: (x + 2) = x  x + 2 = x²
x² - x – 2 = 0
x = [-(-1) ± {(-1)² - 4 × 1 × -2]/2(1) = [1 ± 9]/2 = {2, -1}
Check: 2² - 2 – 2 = 0; (-1)² - (-1) – 2 = 0
3)
The volume of a cube is given by V = s3, where s is the length of a side. Find the length of a
side of a cube if the volume is 800 cm3. Round the answer to three decimal places.
Answer: 9.283 cm
Show your work here:
800 = s³  s = 3800 = 3(100 × 2³) = 2 3100  9.283 cm
4)
For the following function, C computes the cost in millions of dollars of implementing a city
recycling project when x percent of the citizens participate. C ( x) 
a)
1.5 x
100  x
Using this model, approximately what is the cost if 60% of the citizens participate?
Answer: $2.25 million
Show your work here: C(60) = 1.5 × 60/(100 – 60) = 2.25
b)
Using this model, estimate the percentage of participation that can be excepted if $5 million is
spent on this recycling project? Set up an equation and solve algebraically.
Round to the
nearest whole percent.
Answer: 77%
Show your work here:
5 = 1.5 × x/(100 – x)  500 – 5x = 1.5x  6.5x = 500  x = 77
5)
a)
If y  x  2 , fill in the following table for x = 0, 1, 2, 3, 4. Round to three decimal
places where necessary.
x
0
1
2
3
4
y
2
3
3.414
3.732
4
Show your work here:
Substitute the given values in place of x and compute y.
b)
Explain why no negative values are chosen as values to substitute in for x.
Answer: The domain of the x function is [0, ∞) and therefore no negative values are admissible.
c)
Graph in MS Excel and paste your graph here.
Answer:
4)
The formula for calculating the amount of money returned for an initial deposit into a bank
account or CD (certificate of deposit) is given by
nt
r

A  P1  
 n
A is the amount of the return.
P is the principal amount initially deposited.
r is the annual interest rate (expressed as a decimal).
n is the number of compound periods in one year.
t is the number of years.
Carry all calculations to six decimals on each intermediate step, then round the final answer to
the nearest cent.
Suppose you deposit $2,000 for 5 years at a rate of 8%.
a)
Calculate the return (A) if the bank compounds annually (n = 1). Round your answer to the
hundredth's place.
Answer: $2938.66
Show work in this space. Use ^ to indicate the power or use the Equation Editor in MS Word.
A = 2000[1 + 0.08]5 = $2938.66
b)
Calculate the return (A) if the bank compounds quarterly (n = 4). Round your answer to the
hundredth's place.
Answer: $2971.89
Show work in this space:
A = 2000[1 + (0.08/4)]^(5 x 4) = $2971.89
c)
If a bank compounds continuously, then the formula used is A  Pe
where e is a constant and equals approximately 2.7183.
Calculate A with continuous compounding. Round your answer to the hundredth's place.
rt
Answer: $2983.65
Show work in this space:
A = 2000 × e 0.08 × 5 = $2983.65
7)
A commonly asked question is, “How long will it take to double my money?” At 8% interest rate
and continuous compounding, what is the answer? Round your answer to the hundredth's
place.
Answer: 8.66 years
Show work in this space:
A = Pert  A/P = ert, r = 0.08
Here, A/P = 2  e0.08t = 2. On solving, we get t = 8.66 years as the doubling time.
8)
Suppose that the function P  13  45 ln x represents the percentage of inbound e-mail in the
U.S. that is considered spam, where x is the number of years after 2000.
Carry all calculations to six decimals on each intermediate step when necessary.
a)
Use this model to approximate the percentage of spam in the year 2003.
Answer: 62.437553
Show your work in this space:
P = 13 + 45 ln 3 = 62.437553
b)
Use this model to approximate the year that the percent of spam will reach 95% provided that
law enforcement regarding spammers does not change.
Answer: 6.185589  in the 7 th year, that is 2007
Show your work in this space:
95 = 13 + 45 ln x  ln x = (95 – 13)/45 = 1.822222  x = e1.822222 = 6.185589
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