Math 116 — Practice for Exam 2
Generated March 8, 2023
Name:
SOLUTIONS
Instructor: Section Number:
1. This exam has 3 questions. Note that the problems are not of equal difficulty, so you may want to skip
over and return to a problem on which you are stuck.
2. Do not separate the pages of the exam. If any pages do become separated, write your name on them
and p oi nt them out to your instructor when you hand in t he exam.
3. Please read the instructions for each individual exercise carefully. One of the skills being tested on
this exam is your ability to interpret questions, so instructors will not answer questions about exam
problems during the exam.
4. Show an appropriate amount of work (including appropriate explanation) f or each exercise so that the
graders can see not only the answer but also how you obtained it. Include units in your answers where
appropriate.
5. You may use any calculator except a TI-92 (or other calculator with a full alphanumeric keypad).
However, you must show work for any calculation which we have learned how to do in this course. You
are also allowed two sides of a 3
′′
× 5
′′
note card.
6. If you use graphs or tables to ob t ain an answer, be certain to include an explanat ion and sketch of the
graph, and to write out the entries of the table that you use.
7. You must use the methods learned in this course to solve all problems.
Semester Exam Problem Name Points Score
Winter 2022 2 5 data storage 12
Fall 2021 2 3 sheep 14
Winter 2021 2 2 log pile 12
Total 38
Recommended time (based on points): 34 minutes
Math 116 / Exam 2 (March 21, 2022 ) page 7
5. [12 points] A tech startup is gr owing quickly, and the company needs to understand its
customers data-storage needs to properly scale its inf ras t ru ct u r e. Over the course of each
month, the users each store 5 gigabytes of new data. Additionally, because use r s are conscious
of their digital footprint, at the beginning of each month, each u ser deletes 20% of all data
they had stored in previous months.
a. [4 points] Let D
n
be the amount of data stored per user at the end of the n
th
month. If
D
1
= 5, write expre ss ion s for D
2
and D
3
. The letter D should not appear in your final
answers.
D
2
=
5 + 5 (.8)
D
3
=
5 + 5 (.8) + 5 (.8)
2
b. [4 points] Find a closed form expression for D
n
. This means your answer should be a
function of n, should not contain Σ, and sh oul d not be recursive.
D
n
=
5(1 − (.8)
n
)
1 − (.8)
c. [4 points] What is the long-term expected data storage of a user in gigabytes?
Answer =
5
1 − .8
= 25
Math 116 / Exam 2 (November 8, 2021 ) page 3
3. [14 points] Molly has recently become a sheep herder. She rotates her sheep through various
fields so that the sh eep have a varied diet and the fields have a chance to grow. Every Monday,
the sheep visit the same field. Before the sheep graze for the first time in this field, its grass
is 20 centimeters tall. Molly’s sheep are pi cky and only e at the top 40% of the length of grass
in this field every Monday. Over the course of the week, before the next Monday, the gr ass
grows 3 centimeters. Let G
i
represent the height in centimeters of the grass right before the
sheep graze on it for the ith time. Note that G
1
= 20.
a. [5 points] Find expression s for each of G
2
, G
3
, and G
4
. You do not need to evaluate your
expressions.
Solution:
G
2
= (0.6)G
1
+ 3
= (0.6)(20) + 3
G
3
= (0.6)G
2
+ 3
= (0.6)
2
(20) + (0.6)(3) + 3
G
4
= (0.6)G
3
+ 3
= (0.6)
3
(20) + (0.6)
2
(3) + (0.6)(3) + 3
b. [5 points] Find a general closed-form expression for G
n
, defined for n = 2, 3, 4 . . .
Solution:
G
n
= (0.6)
n−1
(20) +
n−2
X
i=0
3(0.6)
i
= (0.6)
n−1
(20) +
3(1 − (0.6)
n−1
)
1 − 0.6
c. [4 points] In order for the field to meet sheep grazing standards, the height of the grass
must be at least 5 cm when the sheep begin grazing. Molly thinks she will be able to stay
on her field forever. Help her determine whether she can stay by either showing that the
grass will eventually be less than 5 cm in height, or sh owing t hat the grass will be at least
5 cm each time before the sheep graze.
Solution:
lim
n→∞
G
n
=
3
1 − 0.6
= 7.5.
Also note that G
n
is a decreasing sequence. So, the grass is always taller than 5 cm.
when the sheep begin grazing.
Math 116 / Exam 1 (March 29, 2021) page 3
2. [12 points] In order to bui l d a sett le ment on the island, intruders start cu t t ing down tree s at
the forest, cutting the trees into logs, and putting the logs in a pile. Let A
n
be the number of
logs they have in the pile at noon on the n-th day. The intruders have 100 logs in the pile at
noon on the first day (so A
1
= 100). Every day (between noon on one day and noon on the
next day), the building team uses 10% of the logs in the pil e, while the log-cutting team adds
20 logs to the pile immediately before noon.
a. [4 points] Find A
2
and A
3
. You do not need to simplify your answers.
Solution:
A
2
= 100 · 0.9 + 20
A
3
= (100 · 0.9 + 20) · 0.9 + 20 = 100 · 0.9
2
+ 20 + 20 · 0.9
b. [5 points] Find a closed form expression for A
n
. Closed form means your answer should
not include ellipses or sigma notation, and should NOT be recursive. You do not need
to simplify your closed form answer.
Solution: From observing the pattern from A
1
, A
2
and A
3
, we have
A
n
= 100 · 0.9
n−1
+ (20 + 20 · 0.9 + 20 · 0.9
2
+ · · · + 20 · 0.9
n−2
)
= 100 · 0.9
n−1
+ 20 ·
1 − 0.9
n−1
1 − 0.9
.
Note that the term 100 · 0.9
n−1
is not part of the geometric series. There are n − 1 te rm s
in the geometric series, so the exponent in the closed form is n − 1.
c. [3 points] How many logs will the intruders have in the pile in the long run?
Solution:
lim
n→∞
100 · 0.9
n−1
+ 20 ·
1 − 0.9
n−1
1 − 0.9
= 0 + 20 ·
1
1 − 0.9
= 20 · 10 = 200.
Winter, 2021 Math 116 Exam 2 Problem 2 (log pile) Solution
