Question

About 10.8 million gallons of oil were spilled into Alaska's Prince William Sound from the 1989 Exxon Valdez oil spill. A major cleanup effort lasted for three years and removed about 99% of the oil from the water and surrounding beaches. However, a 2007 study estimated that about 70,000 gallons of oil were still in the water and on the beaches and that the amount of oil remaining was decreasing at an annual rate of 4%.

(a)

Write an exponential function expressing the amount of oil in the water

O

from 2007 on as a function of time, with

t = 0

in 2007.

O =

(b)

Use your function to estimate, in gallons, the amount of oil that will remain in the water and on the beaches in 2025. (Round your answer to the nearest integer.)

gal

Use your function to estimate, in gallons, the amount of oil that will remain in the water and on the beaches in 2040. (Round your answer to the nearest integer.)

gal

(c)

Some environmentalists feel that the spillage will be effectively "gone" once the amount of oil remaining is less than 1000 gallons. How long will this take in years, based on your model? (Round your answer to the nearest integer.)

yr

(d)

Assuming that the same decay rate has been in effect since the major cleanup efforts ended in 1992, estimate how much oil, in gallons, was in the water and on the beaches at that time. (Round your answer to the nearest integer.)

gal

Answer 1

In   2007 study estimated that about 70,000 gallons of oil were still in the water and on the beaches and that the amount of oil remaining was decreasing at an annual rate of 4%.

That is oil in the next year is 96% of last year.

ie, in 2008 O=70000 *(96/100)^(2008 - 2007)

in 2009 O=70000 *(96/100)^(2009 - 2007)

and so on

ans(a)

O=70000*(96/100)^(2007-2007)

O=70000*(96/100)^0

=70000*1

=70000

ans (b)

In 2025 O=70000*(96/100)^(2025- 2007)

=70000*(96/100)^18

=70000* 0.47960334

=33572.23

=33572

in 2040 O= 70000*(96/100)^(2040-2007)

=70000*(96/100)^33

=70000* 0.25998644

=18199.0508

=18199

ans(c )Some environmentalists feel that the spillage will be effectively "gone" once the amount of oil remaining is less than 1000 gallons. How long will this take in years, based on your model? (Round your answer to the nearest integer.)

ie,O <= 1000

1000<=70000* (96/100)^x

1000/70000<=(96/100)^x

0.01428571<=(96/100)^x

the greatest integer satisfing this is 105.

therefore it will take 105 years

ans(d)

Assuming that the same decay rate has been in effect since the major cleanup efforts ended in 1992, estimate how much oil, in gallons, was in the water and on the beaches at that time. (Round your answer to the nearest integer.)

gal

O=10800000*(96/100)^(1992-1989)

=10800000*(96/100)^3

=10800000*0.884736

=9555148.8

=9555149 gallons