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
water and swims to retrieve the ball. The ball enters the water 200 feet down
from where they stand and 50 feet from the shore. Hunter can run 8.8 feet per
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
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.
(c) Use your properties of logarithms to write equivalent
expressions for each of the
following. Simplify your expression whenever possible. Each equivalent
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
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
growing at a continuous rate of 34.66% each year. Thus the chickadee population
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 5-year period.
(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
c) , and
are equivalent expressions. True False
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