Question

1.

The total world population is forecast to be

P(t) = 0.00073t³ − 0.072t² + 0.9t + 6.04    (0 ≤ t ≤ 10)

in year t, where t is measured in decades, with t = 0 corresponding to 2000 and P(t) is measured in billions.

a. World population is forecast to peak in what year? Hint: Use the quadratic formula. (Remember that t is in decades and not in years. Round your answer down to the nearest year.)

________

b. At what number will the population peak? (Be sure to use the value of t found in part (a). Round your answer to two decimal places.)

________ billion

2.  Postal regulations specify that a parcel sent by priority mail may have a combined length and girth of no more than 150 in. Find the dimensions of a rectangular package that has a square cross section and largest volume that may be sent by priority mail. (Hint: The length plus the girth is 4x + l.)

a. length = 150/3 in

b. width = ___ in

c. height = ___ in

d. What is the volume of such a package?

_____ in^3

3.

A introduced the first iPod in October 2001. Sales of the portable music player grew slowly in the early years but began to grow rapidly after 2005. But the iPod era is coming to a close. Smartphones with music and video players are replacing the iPod, along with the category of device it helped to create. Sales of the iPod worldwide from 2007 through 2011 (in millions) were approximately

N(t) = −2.65t² + 13.13t + 39.9    (0 ≤ t ≤ 4)

a. in year t, where t = 0 corresponds to 2007.† Show that the worldwide sales of the iPod peaked sometime in 2009 by finding the time t where N(t) has a maximum. (Round your answer to two decimal places.)

t = ___

b. What was the approximate largest number (in millions of units) of iPods sold worldwide from 2007 through 2011? (Round your answer to one decimal place.)

____ million units

4.

Two chemicals, A and B, interact to form a Chemical C. Suppose the amount (in grams) of Chemical C formed t min after the interaction begins is

A(t) =

170(1 − e^0.022662t)

1 − 2.5e^0.022662t

(a).How fast is Chemical C being formed 1 min after the interaction first began? (Round your answer to two decimal places.)

_____ g/min

(b)How much Chemical C will there be eventually? Hint: Evaluate lim t→∞ A(t).

______ g

5.

Find the rate of change of y with respect to x at the indicated value of x.

y =

x tan(x)

5 sec(x)

;    x = 0

y'=________

6. A major network is televising the launching of a rocket. A camera tracking the liftoff of the rocket is located at point A, as shown in the accompanying figure, where ϕ (phi) denotes the angle of elevation of the camera at A. How fast (in rad/sec) is ϕ changing at the instant when the rocket is at a distance of 13,000 ft from the camera and this distance is increasing at the rate of 450 ft/sec? (Round your answer to three decimal places.)

line segment = 12,000 ft

____ rad/sec

7.  The accompanying figure depicts a racetrack with ends that are semicircular in shape. The length of the track is 1056 ft (1/5 mi). Find l and r such that the area of the rectangular region of the racetrack is as large as possible. (Round your answers to the nearest foot.)

r = ____ ft

l = ____ ft

What is the area enclosed by the track in this case? (Round your answer to the nearest square foot.)

____ ft^2

Answer 1