Find the associated exponential growth or decay model.
a) = 2000 when = 0; Half-life = 5.
Solution: This part says half-life, so we are dealing with
exponential decay. Therefore we use the formula
ln 2
ln 2
() = 0 , where =
=
.
half-life
5
Also, 0 is def
The half-life of strontium-90 is 28 years.
a) Obtain an exponential decay model for strontium-90 in the form
() = 0 .
ln(2)
Solution: Use the formula =
to find : =
half-life
formula is
() = 0 0.0248 .
ln(2)
28
0.0248. The
b) Use the model to predict, t
Convert the given exponential function to the form indicated.
a) ( ) = 2.1 0.1 ; ( ) =
Solution: The given function has the form ( ) = 2.1( 0.1 )
and we want to write it in the form ( ) = 2.1() . For these
to be equal, the quantities in the parentheses m
A bacteria culture starts with 1,000 bacteria and doubles in size
every 3 hours. Find an exponential model for the size of the
culture as a function of time t in hours.
Solution: Use the formula = (example 4 on p. 636 is
similar). The starting amount of b
Find and simplify
(+)()
for the following functions:
a) ( ) = 2 + 3
b) ( ) = 2 +
Solutions: For both parts, you are given a formula and you need to plug your function
into the formula. This requires that you find ( + ) for two different functions. The
pr
Given ( ) = 2 2 + 1, find:
a) (3)
b) ( + )
Solutions: This problem is similar to #7.
(3) = 2(3)2 (3) + 1 = 2(9) + 3 + 1 = 22
( + ) = 2( + )2 ( + ) + 1.
If you want to expand part b) further, remember that
( + )2 = ( + )( + ) = 2 + 2 + 2 . So, the answer
f
Sketch the graph, and find the domain of the following functions:
1
2
3
= ,
= ,
= ,
= ,
= ,
= ,
= | |,
=5
Solutions:
The first graph is here. Its domain is all real numbers, or (, ). Remember that the domain is
the set of all possible input values. An
The following graph shows an index () of productivity in the
US, where is the time in years and = 0 represents January
2000.
a) What is the domain of ?
b) Estimate (0.5), (0) and
(1.5). Interpret your
answers.
Solutions:
Recall that the domain is the set
Given ( ) = 3 + 4, find:
a) (1)
b) (2)
c) ( + )
Solutions:
Whenever you want to find (something), you plug the
something into the function wherever you see an . So,
(1) = 3(1) + 4 = 3 + 4 = 7,
(2) = 3(2) + 4 = 6 + 4 = 2
( + ) = 3( + ) + 4 = 3 3 + 4
Use the graph of the function to find approximations of the
given values.
a) (2)
b) (0)
c) (2)
d) (2) (2)
Solutions:
For each part, remember that the number in the parentheses
next to the is the input value, and you want to find the
corresponding output v
Use the graph of the function to find the approximations of the
given values.
a) (1)
b) (0)
c) (1)
d)
(3)(1)
31
Solutions:
This question is similar to the previous one. One difference
here is that this is a piecewise function, so we have to be aware
that
Problem 5. Sketch the graph of the given function, and
evaluate the given expressions.
2 , if 2 < 0
a) = cfw_
, if 0 < < 4
(1), (0), (1)
Solution: The graph is to
the right. The part after
the if tells us which values are valid for that
particular piece
Evaluate or estimate each expression based on the following
table.
3 2 1 0
() 1 2 4 2
1 2
3
1 0.5 0.25
a) (0)
b) (2)
c) (1) (1)
d) (1)(2)
Solutions:
For each part, remember that the quantity in the parentheses
after the is an -value (an input value), and
The latest demand equation for your gaming website,
www.mudbeast.net, is given by
= 400 + 1,200
where is the number of users who log on per month and is
the log-on fee you charge. Your Internet provider bills you as
follows:
Site maintenance fee:
$20 per
A bag contains three red marbles, two green ones, one
fluorescent pink one, two yellow ones, and two orange ones.
Suzan grabs four at random. Find the probabilities of the
indicated events.
a) She gets all the red ones, given that she does not get the
flu
In order to play the Mega Millions Lottery, we need to choose a
ticket with five numbers from the set
, and one
number from the set
. The order of the first five
numbers does not matter.
a) How many different tickets can we buy?
Solution: We can view this
Find the conditional probabilities of the indicated events
when two fair dice (one red and one green) are rolled.
a) The sum is 6, given that the green one is either 4 or 3.
Solution: Let be the event that the sum of the two dice is 6,
and let be the even
Supply the missing
quantities.
Solution: The sum of the probabilities on the branches leaving
any node is always 1 (see p. 501), and since
this
means
(so thats what we put in the bottom left
blank). This allows us to fill in the middle two blanks as well.
Compute the indicated quantity.
a)
Find
Solution: Use the formula
This gives
.
b)
Find
Solution: Use the same formula from part a), but solve for
to get
.
c)
and
are independent. Find
.
Solution: When two events are independent,
Thus,
.
A test has three parts. Part A consists of eight true false
questions, Part B consists of five multiple choice questions with
five choices each, and Part C requires you to match five
questions with five different answers one-to-one. Assuming
that you make
Tyler and Gebriella are among seven contestants from which
four semifinalists are to be selected at random. Find the
probability that neither Tyler nor Gebriella is selected.
Solution: Here
is the total number of ways that you can
choose a group of four p
A packet of gummy candy contains four strawberry gums, four
lime gums, two black current gums, and two orange gums.
April May sticks her hand in and selects four at random.
Complete the following sentences:
a) The sample space is the set of
Solution: The
Suzy is given a bag containing 4 red marbles, 3 green ones, 2
white ones, and 1 purple one. She grabs five of them. Find the
probabilities of the following events, expressing each as a
fraction in lowest terms.
For all of these problems,
is the same. In t
If 10 persons met at a reunion and each person shakes hands
exactly once with each of the others, what is the total number
of handshakes?
Solution: We can think of this in the following way: if we choose
any two people and put them together in a set, they
Describe the sample space of the experiment and list the
elements of the given event.
a) Three coins are tossed; the result is at most one head.
Solution: is the set of all outcomes when three coins are
tossed. Written as a set, this is
The event space is
Ben and Ann are among 7 contestants from which 4
semifinalists are to be selected. Of the different possible
selections, how many contain Ben but not Ann?
Solution: There are 7 contestants, but since Ann is excluded and
Ben must be included, there are onl
A bag contains 3 red marbles, 2 green ones, 1 lavender one, 2
yellows, and 2 orange marbles.
a) How many possible sets of four marbles are there?
Solution: The question can be reworded as how many
ways can you choose four marbles from a set of ten
marbles
How many three-letter (unordered) sets are possible that use
the letters q, u, a, k, e, s at most once each?
Solution: Since it says unordered, we know we must be using
combinations. We would like to find the number of ways to
choose a set of three letter
How many unordered sets are possible that contain three
objects chosen from seven?
Solution: Remember that when choosing between
permutations and combinations, you have to decide if order
matters (permutations) or not (combinations). In this case,
order d
How many five-letter sequences are possible that use the
letters b, o, g, e, y once each?
Solution: You need to find out how many ways you can fill in
five blank spaces _ _ _ _ _ using the five given letters. For the
first blank space there are five choic