a sample of the paint used in a cave painting in france is found to have lost 82% of its original...

0.000121 0.000121

A sample of the paint used in a cave painting in France is found to have lost 82% of its original carbon-14. Solving this equation approximates the number of years ago the painting was made.

0.0001210.18 te

0.0001210.18 te

0.000121ln0.18 ln te

ln0.18 0.000121t

14172 t

0.000121 141720.18

1.7150.18

0.18 0.17996

e

e

The painting was made about 14,000 years ago.

Oper Inv

0.000121 0.000121

^ lne

96 96

0.14 0.14

A cup of coffee contains approximately 96 mg of caffeine. When you drink the coffee, the caffeine is absorbed into the bloodstream and is eventually metabolized by the body. The equation below predicts the hours it takes for the amount of caffeine to be reduced to 12 mg?

0.1412 96 te 0.1412 96 te

0.141296

te

0.1412ln ln96te

12ln 0.1496 t

14.85 t

0.14 14.85

2.079

2.079

12 96

12 96

12 96

12 96 0.125

12 12.005

e

e

e

A little under 15 hours.

Oper Inv

0.14 0.14

^ ln

96 96

e

0.05 0.05

20,000 20,000

At 5% interest compounded continuously the equation below finds how long it will take for $20,000 to become $100,000.

0.05100,000 20,000 te

0.05100,000 20,000 te

0.055 te

0.05ln5 ln te

ln5 0.05t

32.1888 t

0.05 32.1888

1.60944

100,000 20,000

100,000 20,000

100,000 20,000 5

100,000 100,000

e

e

About 32 years

Oper Inv

0.05 0.05

^ ln

20,000 20,000

e

55 55

The profit (loss) an orchestra makes (in $10,000) can be approximated by their lowest ticket price. The equation below predicts the price which will earn a profit of $400,000.

40 55ln 68x

40 55ln 68x 68 68

108 55ln x

108ln

55x

108ln55 xe e

7.13 x

40 5.5ln 4959.6522 6.8

40 5.5 8.5091 6.8

40 46.8 6.8

40 40

When their lowest ticket price is a little over $7 their profit will be approximately $400,000.

Oper Inv

ln ^

55 55

68 68

e

5.5 5.5

Since 1985 the pounds of beef consumed per person per year in the United States has been decreasing. The equation below gives the first year per person consumption will drop to 60 pounds per person per year. (You have to add your answer to 1985 to get the year.)

60 5.5ln 1 79x Oper Inv

1 1

ln ^

5.5 5.5

79 79

e

60 5.5ln 1 79x 79 79

19 5.5ln 1x

19 ln 15.5 x

19

5.5 ln 1xe e

195.5

1e x

1 1

195.5

1e x

30.6 x

60 5.5ln 30.6 1 79

60 5.5ln 31.6 79

60 5.5 3.45 79

60 60.025

Around 2015

A rumor is spreading across a campus with 1,500 students. Solving the equation below will approximate the number of students who have heard the rumor after 3 days.

13 6.5ln 1

1500x

Oper Inv

1 1500

1500 11 1

ln ^

6.5 6.5

e

13 6.5ln 1

1500x

6.56.5

13 ln 16.5 1500x

1ln 1

15003

6.5 xe e

36.5 1

11500

e x

36.5 1

11500

e x

1 1

36.5 1

11500

e x

36.5 1

1500( 1) 15001500

e x

554.5 x

Around 555 students.