Exponential And Logarithmic Equations Worksheet - Section 4-7 (Page 10 of 11) in pdf

323

4-7 Exponential and Logarithmic Equations

where P is the population after t years, P

is the population

How many days, to the nearest day, will the advertising

at t

0, and the population is assumed to grow continu-

campaign have to last so that 80% of the possible viewers

ously at the annual rate r. How many years, to the nearest

will be aware of the product?

year, will it take the world population to double if it grows

89. Newton’s Law of Cooling. This law states that the rate at

continuously at an annual rate of 2%?

which an object cools is proportional to the difference in

84. World Population. Refer to Problem 83. Starting with a

temperature between the object and its surrounding

world population of 4 billion people and assuming that the

medium. The temperature T of the object t hours later is

population grows continuously at an annual rate of 2%,

given by

how many years, to the nearest year, will it be before there

is only 1 square yard of land per person? Earth contains

where T

is the temperature of the surrounding medium

approximately 1.7

square yards of land.

and T

is the temperature of the object at t

0. Suppose a

— Carbon 14 Dating. As long as a plant or

85. Archaeology—

bottle of wine at a room temperature of 72°F is placed in a

animal is alive, carbon 14 is maintained in a constant

refrigerator at 40°F to cool before a dinner party. After an

amount in its tissues. Once dead, however, the plant or an-

hour the temperature of the wine is found to be 61.5°F.

imal ceases taking in carbon, and carbon 14 diminishes by

Find the constant k, to two decimal places, and the time, to

radioactive decay according to the equation

one decimal place, it will take the wine to cool from 72 to

50°F.

0.000124t

where A is the amount after t years and A

is the amount

90. Marine Biology. Marine life is dependent upon the micro-

when t

0. Estimate the age of a skull uncovered in an

scopic plant life that exists in the photic zone, a zone that

archaeological site if 10% of the original amount of car-

goes to a depth where about 1% of the surface light still

bon 14 is still present. Compute the answer to three signif-

remains. Light intensity is reduced according to the expo-

icant digits.

nential function

— Carbon 14 Dating. Refer to Problem 85.

86. Archaeology—

What is the half-life of carbon 14? That is, how long will

where I is the intensity d feet below the surface and I

it take for half of a sample of carbon 14 to decay? Com-

the intensity at the surface. The constant k is called the

pute the answer to three signiﬁcant digits.

coefﬁcient of extinction. At Crystal Lake in Wisconsin it

was found that half the surface light remained at a depth

87. Photography. An electronic ﬂash unit for a camera is acti-

of 14.3 feet. Find k, and ﬁnd the depth of the photic zone.

vated when a capacitor is discharged through a ﬁlament of

Compute answers to three signiﬁcant digits.

wire. After the ﬂash is triggered and the capacitor is dis-

charged, the circuit (see the ﬁgure) is connected and the

91. Agriculture. Table 1 shows the yield (bushels per acre)

battery pack generates a current to recharge the capacitor.

and the total production (millions of bushels) for corn in

The time it takes for the capacitor to recharge is called the

the United States for selected years since 1950. Let x rep-

recycle time. For a particular ﬂash unit using a 12-volt bat-

resent years since 1900.

tery pack, the charge q, in coulombs, on the capacitor t

seconds after recharging has started is given by

T A B L E

1 United States Corn

0.2t

0.0009(1

)

Production

How many seconds will it take the capacitor to reach a

charge of 0.0007 coulomb? Compute the answer to three

Yield

Total Production

signiﬁcant digits.

Year

(bushels per acre)

(million bushels)

1950

37.6

2,782

1960

55.6

3,479

1970

81.4

4,802

1980

97.7

6,867

1990

115.6

7,802

88. Advertising. A company is trying to expose a new product

to as many people as possible through television advertis-

Source: U.S. Department of Agriculture.

ing in a large metropolitan area with 2 million possible

viewers. A model for the number of people N, in millions,

who are aware of the product after t days of advertising

(A) Find a logarithmic regression model (y

b ln x)

was found to be

for the yield. Estimate (to one decimal place) the yield

0.037t

2(1

)

in 1996 and in 2010.

Exponential And Logarithmic Equations Worksheet - Section 4-7 Page 10

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