9. Mark and his dog Hunter are standing together on the
shore at the beach. Mark throws a ball
into the ocean and Hunter runs along the shoreline until some point in which he
enters the
water and swims to retrieve the ball. The ball enters the water 200 feet down
the shoreline
from where they stand and 50 feet from the shore. Hunter can run 8.8 feet per
second along
the shore but can only swim 3.6 feet per second in the ocean. At what point
along the shore
should Hunter enter the water in order to minimize his total time?
Let x represent the distance from where Hunter enters the water to the point on
the shore
opposite the ball.
(a) Write an expression in terms of x that represents
the distance Hunter will run along the shore. ……………………………
the distance that Hunter swims …………………………………………...
(b) The total time it takes Hunter to reach the ball is given by
where = the distance Hunter runs along the
shore, = the rate at which Hunter runs,
= the distance Hunter swims and
= the rate at which Hunter swims. Write a
function of T in terms of x .
(c) At what point should Hunter enter the water in order to minimize the time it
takes him to
reach the ball? Document your process for answering this question.
10. In 1980, the average price of a house in a certain
county was $80,000. Prices have increased
at an average rate of 5% every three years. Let P(x) represent the price of a
house in this
county x years after 1980.
(a) Complete the table below:
(b) Write an algebraic representation for P(x) using
fractional exponents.
(c) Find the exponential regression model for this data. What does the
regression model
suggest about the growth rate of housing prices in this time period?
(d) How could you have predicted the regression model from your answer to (b)?
(e) How much will a house sell for in 2000? Document your process.
(f) When will a house be worth $150,000? Document your process.
11. Miranda bought a truck for $40,000 with an expected
halflife of 3 years.
(a) Complete the table below.
(b) Write an algebraic model for this problem situation
using fractional exponents. Define
your variables .
(c) Find an exponential regression model for the table above.
(d) What is the annual depreciation of this truck?
(e) Explain why the algebraic model and the regression model are equivalent?
12. The interest formula for compounding n times a year is
given by .
(a) How much money will Jason accrue in 10 years if he deposits $10,000 in an
account
paying 6% compounded monthly?
(b) How long will it take $5000 to double in value if it is deposited into a
certificate of
deposit that pays 4.5% compounded quarterly?
13. (a) Write in
exponential form. Then verify the result using what you know
about fractional exponents ( not your calculator ).
(b) Change from exponential to logarithmic for , or vice versa. Solve for the
variable.
(c) Use your properties of logarithms to write equivalent
expressions for each of the
following. Simplify your expression whenever possible. Each equivalent
expression will
contain either an x or log x and other constants.
14. Biologists often model population growth using the
number e since this is continuous
reproduction in populations in normal circumstances. A group of biologist
studying
chickadee populations in West Texas estimate that currently there are
approximately half a
million chickadees in the region. Based on past studies, they estimate that the
population is
growing at a continuous rate of 34.66% each year. Thus the chickadee population
can be
modeled by the function , where x is the
number of years from now and y
is the Chickadee population.
(a) How long does it take the chickadee population to double?
(b) Make a table that shows the chickadee population for a 5year period.
Years 
0 
1 
2 
3 
4 
5 
Chickadee
Population 






(c) What pattern do you notice in the table? Write another
model for the population growth
based on this pattern.
(d) Explain why the two models you have constructed are equivalent?
15. A bit more skill practice True / False or Fill
in the blank.
a) ln 7 means . True False
b) The expression x(x + 4) + 7 is in factored form . True False
True False
c) , and
are equivalent expressions . True False
d) The opposite of 4 – x is x – 4. True False
e) A rational expression is a fraction whose numerator and denominator can be
factored.
True False
f) is equivalent to
True False
g) The product of any complex number and its conjugate is always a real number .
True False
h) What number must be added to both sides of x^{2} – 14x = 20 to “complete the
square”? …………………….
i) True False
j) True False
k) The expressions and
are equivalent. True False
l) The x coordinate of the vertex of the parabola y = 4x^{2} – 12x + 11 is
……………………
m) True False
n) The expression can be simplified by
subtracting exponents True False
o) The domain of is………………………..
p) The difference of squares A^{2}– B^{2} can be factored as …………………………….
q) True False
r) The expression represents a negative number . True False