Math 116 — Practice for Exam 3

Generated April 23, 2014

Name:

Instructor: Section Number:

1. This exam has 11 questions. Note that the problems are not of equal difficulty, so you may want toskip 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 themand point them out to your instructor when you hand in the exam.

3. Please read the instructions for each individual exercise carefully. One of the skills being tested onthis exam is your ability to interpret questions, so instructors will not answer questions about examproblems during the exam.

4. Show an appropriate amount of work (including appropriate explanation) for each exercise so that thegraders can see not only the answer but also how you obtained it. Include units in your answers whereappropriate.

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. Youare also allowed two sides of a 3′′ × 5′′ note card.

6. If you use graphs or tables to obtain an answer, be certain to include an explanation and sketch of thegraph, 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 2013 3 5 skydiver 14

Winter 2011 1 7 paint truck 11

Fall 2006 1 8 instruction speed 12

Winter 2011 3 7 wine glass 12

Fall 2007 1 3 pyramid 10

Fall 2013 1 1 light bulb 7

Fall 2010 3 2 sewage tank 7

Winter 2011 1 8 sand dune 12

Winter 2009 2 2 submarine 60

Fall 2013 3 9 olive oil 13

Winter 2014 1 7 drag queens 12

Total 170

Recommended time (based on points): 125 minutes

Math 116 / Final (April 26, 2013) page 7

5. [14 points] A skydiver jumps from a plane at a height of 2, 000 meters above the ground.After some time in free-fall, he opens his parachute, reducing his speed, and lands safelyon the ground.

a. [5 points] The graph of the skydiver’s downward velocity v(t) (in meters per second)t seconds after he jumped is shown below.Sketch the graph of the antiderivative y(t) of v(t) satisfying y(0) = 0. Make sureyour graph reflects the regions at which the function is increasing, decreasing, con-cave up or concave down.

b. [3 points] Write down a right-hand sum with 4 subintervals in order to approximatethe average downward velocity of the skydiver during the time the skydiver is infree-fall. Show all the terms in your sum.

c. [2 points] Is your estimate in (b) guaranteed to be an underestimate or overestimateof the average velocity of the skydiver, or there is not enough information to decide?Justify.

d. [4 points] Find a formula for the height H(t) (in meters) above the ground of theskydiver t seconds after he jumped.

University of Michigan Department of Mathematics Winter, 2013 Math 116 Exam 3 Problem 5 (skydiver)

Math 116 / Exam 1 (February 2011) page 9

7. [11 points]A truck carrying a large tank of paint leaves a garage at 9AM. The tank starts to leak in sucha way that x miles from the garage, the density of paint on the road is e−x2/5000 gallons permile. At 10AM, a cleaning crew leaves from the same garage and follows the path of the truck,scrubbing the paint from the road as it travels until it catches up to the leaking truck. At t

hours after 10AM, the leaking truck is 50 ln(t+2) miles from the garage, and the cleanup crewis 35t miles from the garage. You may use your calculator to evaluate any definite integralsfor this problem.

a. [4 points] Calculate the total amount of paint that has leaked from the truck by 11AM.

b. [2 points] At time t hours after 10AM, what interval I of the road is still covered in paint?(you may assume that t represents a time before the trucks meet)

I =

[

,

]

c. [3 points] Let P (t) represent the amount of paint in gallons on the road t hours after 10AM. Find a formula (which may include a definite integral) for P (t).

d. [2 points] Calculate P ′(1).

University of Michigan Department of Mathematics Winter, 2011 Math 116 Exam 1 Problem 7 (paint truck)

Math 116 / Exam 1 (October 11, 2006) page 9

8. [12 points] In class, Chris’ calculus professor is well known to cover material at a rate m(t) =1

12(t−20)2/3textbook sections/minute, where t is the time in minutes since the start of class.

(a) [2 of 12 points] What is the meaning of the integral∫ 80

0m(t) dt (include units in your explanation)?

(b) [4 of 12 points] How many sections would you estimate the professor covers in the first minute ofclass? In the 20th minute? Why?

(c) [6 of 12 points] Find exactly (that is, by hand) the value of∫ 80

0m(t) dt.

University of Michigan Department of Mathematics Fall, 2006 Math 116 Exam 1 Problem 8 (instruction speed)

Math 116 / Final (April 2011) page 9

7. [12 points]

a. [5 points] You rotate the region shown about the y-axis to create a drinking glass.Write an expression that represents the volume of material required to constructthe drinking glass (your answer may contain f(y)).

1

3

-3

x = f(y)

x =(

y3

)1/5

b. [7 points] Consider the vessel shown below. It is filled to a depth of 1 foot of water.Write an integral in terms of y (the distance in ft from the bottom of the vessel)for the work required to pump all the water to the top of the vessel. Water weighs62.4 lbs/ft3.

1 ft.

y

2 ft.

5 ft.

3 ft.

University of Michigan Department of Mathematics Winter, 2011 Math 116 Exam 3 Problem 7 (wine glass)

Math 116 / Exam 1 (October 10, 2007) page 5

3. [10 points] The Great Pyramid of Giza in Egypt was originally (approximately) 480 ft high. Its basewas originally (again, approximately) a square with side lengths 760 ft.

(a) [6 points of 10] Sketch a slice that could be used to calculate the volume of the pyramid by integrating.In your sketch, indicate all variables you are using. Find an expression for the volume of the slicein terms of those variables.

(b) [4 points of 10] Use your slice from (a) to find the volume of the pyramid.

University of Michigan Department of Mathematics Fall, 2007 Math 116 Exam 1 Problem 3 (pyramid)

Math 116 / Exam 1 (October 9, 2013) page 2

1. [7 points] A lightbulb is obtained by revolving the curve x = f(y) around the y-axis:

(a) Graph of x = f(y) (b) 3D view of the bulb

The following table gives values of x = f(y):

y 0 1 2 3 4 5 6

x = f(y) 0.8 1.1 1.5 1.9 2.4 2.8 0

a. [4 points] Write an integral involving f(y) that computes the volume of the lightbulb.

b. [3 points] Estimate the volume using the midpoint rule. Use the largest number ofsubintervals possible, given the information in the table above. Write out each of theterms in the sum.

University of Michigan Department of Mathematics Fall, 2013 Math 116 Exam 1 Problem 1 (light bulb)

Math 116 / Final (December 17, 2010) page 4

2. [7 points] Deep beneath Dennison Hall lies a large septic tank. It has the shape of atriangular prism with the dimensions depicted below.

Suppose that the tank described above is full of sewage and that this sewage has adensity of 1000(1 + e

−2x) kg

m3 , where x is the distance in meters above the base of thetank.

a. [5 points] Find a definite integral that computes the mass of the sewage in the tank.

b. [2 points] Compute the value of the integral using your calculator. Do not forgetto include the units.

University of Michigan Department of Mathematics Fall, 2010 Math 116 Exam 3 Problem 2 (sewage tank)

Math 116 / Exam 1 (February 2011) page 10

8. [12 points] Sand dunes come in many shapes. Barchan dunes, which have the shape shown onthe left, are studied extensively by geomorphologists. Horizontal cross-sections of these dunesare crescent-shaped (the dashed line encloses one such cross-section), and can be approximatedas the shape on the right. The area of this shape is given by the formula Ah = K(π

2Q2−

4

3Q1).

h KQ1

Q2

You are studying a barchan dune of 10 meters height, for which the values of Q1, Q2, and K

vary with respect to the height h (in meters) of the cross-section according to the functionsQ1(h) = 10−h, Q2(h) = 20−2h, K(h) = 100−h2. The density of sand in the dune is δ = 1600kilograms per cubic meter.

a. [5 points] Write an expression for the volume of one slice of sand dune h meters abovethe ground and ∆h meters thick.

b. [5 points] Write a definite integral that represents the total mass of sand in the dune.You do not need to evaluate this integral.

c. [2 points] Write an expression (involving integrals) for the height of the center of mass ofthe sand dune. You do not need to evaluate this integral.

University of Michigan Department of Mathematics Winter, 2011 Math 116 Exam 1 Problem 8 (sand dune)

page 3 of 9 Math 116, Exam 2, March 24th, 2009!

2. (60 points) Several different forces act on a submerged submarine and cause it to rise and/or

fall. In this problem, we will use a simplified model of a submarine to explore some of these

forces. To construct our model, we revolve the graph of

f (x) = 100 + 5e0.5 xe!0.1x 0 " x " 5

105 5 < x " 105

#$%

&%

around y = 100 (see picture below) . Note that all units in the horizontal and vertical

directions are

measured in meters.

We will also assume

that ocean water has

density of 1025kgm3

and that the density of

material inside the

submarine is a constant

represented by the

symbol ! . Additionally, the acceleration due to gravity is

9.8m

sec2∀!!

a. The force due to buoyancy is an upward force equal to the weight of the water displaced

by the volume of the submarine. Find the volume of water displaced by the sub (i.e. the

volume of the submarine) and the resulting force due to the buoyancy.

y=100

5 105 Not drawn to scale

University of Michigan Department of Mathematics Winter, 2009 Math 116 Exam 2 Problem 2 (submarine)

page 4 of 9 Math 116, Exam 2, March 24th, 2009!

b. Find the center of mass of the nose of the submarine (i.e. the shape of the first 5 meters of

our model). Note it is only necessary to set up, but not calculate, (an) integral(s).

!

!

!

!

!

3. (60 points) The following two questions refer to the submarine described in problem #2.

a. The buoyancy properties of the empty submarine described in problem 1 cause the

submarine to begin moving upward through the ocean water. This motion, in conjunction

with the ocean water, creates a damping force that begins to slow the submarine. Assume

that the damping force is proportional to the square of the velocity of the submarine, and

that when the velocity is 5m/s the force is 100N. For our model submarine, the velocity

at t seconds can be described by

v(t) = 25! 25sin" t

60

#$%

&'(

#

$%&

'(

1

3

(in meters per second). Find

the amount of work the damping force does on the submarine over the first 30 seconds of

motion.

University of Michigan Department of Mathematics Winter, 2009 Math 116 Exam 2 Problem 2 (submarine)

Math 116 / Final (December 17, 2013) page 12

9. [13 points] Olive oil has been poured into the Math Department’s starfish aquarium! Theshape of the aquarium is a solid of revolution, obtained by rotating the graph of y = x

4 for0 ≤ x ≤ 1 around the y-axis. Here x and y are measured in meters.

The aquarium contains water up to a level of y = 0.6 meters. There is a layer of oil of thickness0.2 meters floating on top of the water. The water and olive oil have densities 1000 and 800kg per m3, respectively. Use the value of g = 9.8 m per s2 for the acceleration due to gravity.

a. [6 points] Give an expression involving definite integrals that computes the total mass ofthe water in the aquarium.

b. [7 points] Give an expression involving definite integrals that computes the work necessaryto pump all the olive oil to the top of the aquarium.

University of Michigan Department of Mathematics Fall, 2013 Math 116 Exam 3 Problem 9

Math 116 / Exam 1 (February 10, 2014) page 8

7. [8 points] Alyssa Edwards wants to play a prank on Coco Montrese by spilling a bucket oforange cheese powder on her. To do this Alyssa lifts the bucket at a constant speed from theground to a height of 10 meters. Unfortunately the bucket has a small hole and the cheesebegins leaking out at a constant rate as soon as the bucket leaves the ground. The bucketinitially weighs 10kg and when it reaches a height of 10 meters it only weighs 5kg. Recall thegravitational constant is g = 9.8m/s2.

a. [3 points] Write an expression giving the mass of the bucket m(h) when the bucket is hmeters above the ground.

b. [5 points] How much work is required to lift the bucket from the ground to a height of 10meters? Include units.

University of Michigan Department of Mathematics Winter, 2014 Math 116 Exam 1 Problem 7 (drag queens)