B Advertising Costs The cost, in thousands of dollars, of airing 30 -second television commercials during a Super Bowl game can be

approximated by C(x)=20+4,000x+0.05x2. How many 30-second television commercials should your company air to

minimize average costs? What is the resulting average cost of a 30-second ad?

B Revenue: Smartphones Worldwide annual sales of smartphones in 2012–2013 were approximately q= -6p+3,030 million phones at a selling price

of $p per phone. What selling price would have resulted in the largest annual revenue? What, to the nearest $10 million,

would have been the resulting annual revenue? (The actual selling price in 2013 was $335.)

C Carry-On Dimensions Fly-By-Night Airlines requires that the total outside dimensions (length +width + height) of a carry-on bag not exceed 45

inches. Suppose you want to carry on a bag whose length is twice its height. What is the largest volume bag of this shape

that you can carry on a Fly-By-Night flight?

C Package Dimensions The U.S. Postal Service (USPS) will accept packages only if the length plus girth is no more than 108 inches. Assuming

that the front face of the package (as shown in the figure) is square, what is the largest volume package that the USPS will

accept?

D Research and Development Spending on research and development by drug companies in the United States t years after 1970 can

be modeled by S(t)= 2.5e0.08t billion dollars. The number of new drugs approved by the Federal Drug Administration

(FDA) over the same period can be modeled by D(t)=10+t drugs per year. When was the function D(t)

S(t) at a maximum?

What is the maximum value of D(t)

S(t)? What does the answer tell you about the cost of developing new drugs? (0 ≤ t ≤ 50)

E Marketing Strategy FeatureRich Software Company has a dilemma. Its new program, Doors-X 10.27.104.a, is almost ready to go on the

market. However, the longer the company works on it, the better it can make the program and the more it can charge for

it. The company’s marketing analysts estimate that if it delays t days, it can set the price at 100+2t dollars. On the other

hand, the longer it delays, the more market share it will lose to its main competitor, Orange LLC., so if it delays t days it

will be able to sell 400,000-2,500t copies of the program. How many days should FeatureRich delay the release to get the

greatest revenue?

B Incarceration Rate The incarceration rate (the number of persons in prison per residents) t years since 1990 in the United States can be

approximated by 𝑃(𝑡) = 0.04𝑡3 − 2𝑡2 + 40𝑡 + 460; (0 ≤ 𝑡 ≤ 28). When, to the nearest year, was the incarceration rate

increasing most rapidly? When was it increasing least rapidly?

E Minimum Cost An industrial tank formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the

tank is 4,000 cu. ft. The hemispherical ends cost twice as much per square foot of surface area as the lateral side. Find the

dimensions that will minimize cost.

F Profit Analysis A manufacturer can sell x units per week at a price of P(t) = 200 - 0.01x cents and to produce x units it costs

C(t) = 0.50x + 200 dollars. What manufacturing quantity maximizes profit and what price should the units be sold at?

A Matching Functions to Data From the data in the table below, determine a possible formula for each function. Write an explanation of your reasoning.

x f(x) g(x) h(x) k(x) m(x) n(x)

-5 -10 20 25 0.95892 -0.83907 2.19056

-4.5 -9 19 20.25 0.97753 -0.91113 1.85888

-4 -8 18 16 0.75680 -0.14550 1.54272

-3.5 -7 17 12.25 0.35078 0.75390 1.27691

-3 -6 16 9 -0.14112 0.96017 1.09070

-2.5 -5 15 6.25 -0.59847 0.28366 1.00459

-2 -4 14 4 -0.90930 -0.65364 1.02806

-1.5 -3 13 2.25 -0.99749 -0.98999 1.15852

-1 -2 12 1 -0.84147 -0.41615 1.38163

-0.5 -1 11 0.25 -0.47943 0.54030 1.67280

0 0 10 0 0.00000 1.00000 2.00000

0.5 1 9 0.25 0.47942 0.54030 2.32719

1 2 8 1 0.84147 -0.41615 2.61837

1.5 3 7 2.25 0.99749 -0.98999 2.84147

2 4 6 4 0.90929 -0.65364 2.97193

2.5 5 5 6.25 0.59847 0.28366 2.99540

3 6 4 9 0.14112 0.96017 2.90929

3.5 7 3 12.25 -0.35078 0.75390 2.72308

4 8 2 16 -0.75680 -0.14550 2.45727

4.5 9 1 20.25 -0.97753 -0.91113 2.14112

5 10 0 25 -0.95892 -0.83907 1.80943

B Fluid Flow The emergent velocity v of a liquid flowing from a hole in the bottom of a tank is given by:

𝑣 = √2𝑔ℎ

where g is the acceleration due to gravity (32 feet per second per second) and h is the depth of the liquid in the tank. Find

the rates of change of velocity with respect to height of the liquid when the height is 9 feet.

B Stopping Distance A car is travelling at a rate of 72 feet per second (49 mph) when the brakes are applied. The position function for the car

is:

𝑠(𝑡) = −1.5t3 + 72𝑡

Use this function to complete the table and find the average velocity, acceleration and jerk for each time period. Graph

the four functions on the same graph and explain the physical meaning of each graph.

0 seconds 1 second 2 seconds 3 seconds 4 seconds Units Average

s(t)

v(t)

a(t)

j(t)

C Evaporation As a raindrop falls, it forms a mostly spherical shape. When it reaches a layer of dry air, it begins to evaporate at a rate

that is proportional to its surface area. Show the radius of the raindrop decreases at a constant rate.

D Ecological Disaster A ruptured oil tanker caused a circular oil slick on the surface of the ocean. When its radius is 150 meters, the radius of

the slick is expanding by 0.1 meter/minute and its thickness is 0.02 meter. At that moment how fast is the area of the slick

expanding? The circular slick has the same thickness everywhere, and the volume of oil spilled remains fixed. How fast is

the thickness of the slick decreasing?

C Roadway Design In order for cars in a curve of a constant radius r not to rely on friction of the tires alone to overcome centrifugal force, the

road is banked at an angle of magnitude θ from the horizontal. The banking angle must satisfy the equation:

𝑟𝑔𝑡𝑎𝑛(𝜃) = 𝑣2

where v is the velocity of the car and g = 32 feet per second per second is the acceleration due to gravity. What is the

relationship between the rate of change in velocity and the rate in change of θ

B Electricity The combined electrical resistance R of a circuit with resistance R1 and R2 connected in parallel is given by:

1

𝑅=

1

𝑅1

+1

𝑅2

Where R, R1, and R2 are measured in ohms. R1 is increasing at a rate of 1 ohms per second and R2 is increasing at 1.5 ohms

per second. At what rate is R changing when R1 = 50 ohms and R2 = 75 ohms?

C Air Traffic Control An airplane is flying at an altitude of 5 miles and passes directly over a radar antenna. When the plane is 10 miles away (s

= 10), the radar detects the distance s is changing at a rate of 240 mph. What is the velocity of the plane?

C Baseball A baseball diamond has the shape of a square with sides 90 feet long. A player stealing third base at a velocity of 25 feet

per second is 20 feet from third base. At what rate is the players distance from home plate changing?

D Shadow Length A 6 foot tall man is walking away from a street light that is 15 feet above the ground at 5 feet per second. When he is 10

feet from the light, how fast is the tip of his shadow moving?

D Angle of Elevation A fish is reeled in at a rate of one foot per second from a point ten feet above the water. At what rate is the angle (θ)

between the line and water changing when there is a total of twenty-five feet of line from the end of the pole to the water?

E Linear vs Angular Velocity A wheel of radius 30 centimeters revolves at a rate of 10 revolutions per second. A dot is painted at a point P on the rim

of the wheel. When is the rate of change of x greatest, and when is it least?

D Moving Shadow I A ball is dropped from a height of 20 meters, 12 meters from the top of a 20 meter lamppost. The ball’s shadow, caused

by the light at the top of the lamppost, is moving along the level ground. How fast is the shadow moving one second

after the ball is released?

A Hours of Daylight as a Function of Latitude Let S(x) be the number of sunlight hours on a cloudless June 21, as a function of latitude, x, Measured in degrees. Racine

(43°43’) has 9 hours, 10 minutes of sunlight on that day. Let 𝑥0 be the latitude of the Arctic Circle (𝑥0 ~ 66°30'). In the

northern hemisphere, S(x) is given, for some constants a and b, by the formula:

𝑆(𝑥) = {𝑎 + 𝑏𝑎𝑟𝑐𝑠𝑖𝑛 (

tan (𝑥)

tan (𝑥0)) 0 ≤ 𝑥 < 𝑥0

24 𝑥0 < 𝑥 < 90

Find a and b so that S(x) is continuous, then graph S(x), for 0° ≤ x < 90°. Does S(x) appear to be differentiable?

E Linear and Angular Velocity A rotating beacon is located one kilometer off a straight shoreline. If the beacon rotates at a reate of 3 revolutions per

minute, how fast (in km h-1) does the beam of light appear to be moving to a viewer who is 0.5 kilometers down the

shoreline?

D Moving Shadow II A sandbag is dropped from a balloon at a height of 60 meters when the angle of elevation to the sun is 30°. Find the rate

at which the shadow of the sandbag is travelling along the ground when the sandbag has fallen 25 meters.

F Velocity of a Piston In the engine below, a 7 inch connecting rod is fastened to a crank shaft of radius three inches. The crankshaft rotates at a

constant rate of 200 revolutions per minute. Find the velocity of the piston when 𝜃 = 𝜋

3.

C Descending Elevator You are situated 300 feet from the base of Tower Glitz Plaza watching an external elevator descend the side of the

building. At a certain instant the elevator is 500 feet away from you, and its distance from you is decreasing at a rate

of 16 ft/sec. How fast is the elevator descending at that instant?

F Ships and our Harbor The S.V. City of New Orleans is sailing towards Racine Harbor at 10 knots, while the T.B. Alexandra Zweiflehopher

is sailing from Racine Harbor at 30 knots. The angle formed by the two boats with the vertex at the harbor is 120°.

How fast is the distance between the two vessels changing at the instant that the S.V City of New Orleans is 8 nm

from the harbor and the T.B. Alexandra Zweiflehopher is 6 nm from the harbor?

nm = nautical mile (approximately 1.1508 miles)

knot = nautical mile per hour