Worsheet #9 Log Word Problems
Directions: solve each problem
1. The Diaz family bought a new house 10 years ago for $80,000. The house is now worth $140,000. Assuming a steady rate of growth, what was the yearly rate of appreciation?
2. A rumor can spread very quickly. The number of people H who have heard a rumor can be approximated by the equation: [pic] where P is the total population, S is the number of people who start the rumor and t is the time in minutes. During lunch two students decide to start a rumor that the principal will let the students out of school one hour early that day. If there are 1600 students in the school, how much time will pass before half of the students have heard the rumor?
3. Mr. and Mrs. Sawyer bought a condominium for $75,000. Assuming that its value will appreciate 6% per year, how much will the condo be worth in 5 years when the Sawyers are ready to move?
4. A radioisotope is used as a power source for a satellite. The power output is given by the equation: [pic],where P is the power in watts and t is the time in days.
a. Find the power available after 100 days.
b. Ten watts of power are required to operate the equipment in the satellite. How long can the satellite continue to operate?
5. Paleontologists study life of past geological periods by studying fossil remains. They use carbon–14
[pic]to estimate the age of fossils. Carbon-14 decays with time. In 5760 years, just half of the mass of this substance will remain. This period is called its half-life. To fine the age of a fossil with just 1/5 of this carbgon-14 remaining paleontologists must use the decay formula for this substance. Find the constant and then determine the age of a fossil that has 1/5 of its carbon-14 remaining.
6. How long will it take for an amount of money invested at an annual rate of 5.75% compounded continuously to triple in value?
7. The monthly payment M on a mortgage is given by: [pic], where r is the monthly rate of interest, P is the principal, and n is the number of months in the life of the mortgage. If a mortgage at 12% annual interest on a principal of $62,000 has a monthly payment of $653, what is the life of the mortgage?
8. Common logarithms are used in the measure of sound. The loudness L in decibels, of a particular sound is defined as [pic], where I is the intensity of the sound and [pic] is the minimum intensity of sound detectable by the human ear. Soft recorded music is about 4000 times the minimum intensity of sound detectable by the human ear. Find the loudness in decibels.
9. Mr. and Mrs. Franco are planning to take a cruise for their twenty-fifth wedding anniversary. They have 6 years to save $5000 for the cruise. If the six-year certificate of deposit they buy now pays 8% interest compounded continuously, how much should the3y invest now in order to have $5000 for the cruise?
10. The annual amount that we spend to attend sporting events can be modeled by:
[pic] where x represents the number of years after 1984 and f(x) represents the total annual expenditures for admission to spectator sports, in billions of dollars. In 2000, approximately how much was spent on admission to spectator sports?
11. Students in a psychology class took a final exam. As part of an experiment to see how much of the course content they remembered over time, they took equivalent forms of the exam in monthly intervals thereafter. The average score for the group, f(t) after t months was modeled by the function:
[pic].
a. What was the average score on the original exam?
b. What was the average score after 2 months,? 6 Months? 10 months? 1 year?
12. Medical research indicates that the risk of having a car accident increases exponentially as the concentration of alcohol in the blood increases. The risk is modeled by: [pic], where x is the blood alcohol concentration and R, given as a percent, is the risk of having a car accident. What blood alcohol concentration corresponds to a 17% risk of a car accident?
13. The exponential growth model [pic] describes the population of the United States, A , in millions, t years after 1970. What was the population of the United States in 1970? When will the U.S. population be 300 million? When will i9t be 350 million?
14. The exponential decay model for carbon-14 is [pic].
a. Prehistoric cave paintings were discovered in a cave in France. The paint contained 15% of the original carbon 14. Estimate the age of the paintings.
b. Skeletons were found at a construction site in San Francisco in 1989. The skeletons contained 88% of the expected amount of carbon-14 found in a living person. In 19879, how old were the skeletons?
c. Bones of a prehistoric man were discovered and contained 5% of the original amount of carbon-14. How long ago did the man die?
15. A bird species in danger of extinction has a population that is decreasing exponentially. Five years ago the population was at 1400 and today only 1000 of the birds are alive. Once the population drops below 100, the situation will be irreversible. When will this happen?
16. The formula [pic] models the population of Los Angeles, California. A, in millions, t years after 1992. If the growth rate continues into the future, when will the population reach 13 million?
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