Practice Makes Perfect Algebra - Carolyn Wheater [30]
If the interest is compounded more than once a year, you don’t get the whole year’s worth of interest at every interval. Instead, the annual rate is divided by the number of times per year the calculation is done. While this might look like it would reduce the interest, remember that interest will be compounded more frequently and each time it is compounded, the amount of money earning interest grows.
For compound interest, use the formula , where P is the principal or original investment, r is the rate of interest per year(converted to a decimal), n is the number of times per year interest is compounded, and t is the number of years.
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EXERCISE 12.1
Calculate the value of each investment after the specified time, when invested as described.
1. An investment of $5000, at 3% per year, compounded annually, for 2 years
2. An investment of $10,000, at 8% per year, compounded semiannually, for 5 years
3. An investment of $2000, at 4% per year, compounded quarterly, for 10 years
4. An investment of $2500, at 5% per year, compounded monthly, for 4 years
5. An investment of $100,000, at 10% per year, compounded quarterly, for 12 years
6. An investment of $4000, at 5.5% per year, compounded monthly, for 8 years
7. An investment of $7500, at 2.5% per year, compounded semiannually, for 5 years
8. An investment of $3000, at 9% per year, compounded annually, for 8 years
9. An investment of $15,000, at 12% per year, compounded quarterly, for 6 years
10. An investment of $25,000, at 6% per year, compounded annually, for 15 years
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Exponential growth and decay
Compound interest is a common example of exponential growth. The principal amount grows over time because it’s multiplied by powers of 1 + r or . When the base of a power is greater than 1, the power will grow as the exponent increases, so equations of the form y = abx represent exponential growth when b is greater than 1.
In contrast, when b is less than 1 but greater than 0, the equation represents exponential decay, a decreasing quantity. If the amount of math George knows decreases 2% per week over summer vacation, after 1 week he’ll know 98%(or 1 − 0.02) of what he knew the week before. After 3 weeks, he’ll know 98% of 98% of 98%, or(1 − 0.02)3. When the rate of increase or decrease is given, it may be helpful to rewrite the equation as y = a(1 + r)x for increase or y = a(1 − r)x for decrease.
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EXERCISE 12.2
Tell whether each equation represents exponential growth or exponential decay.
1. y = 400(1 + 0.03)x
2. y = 30,000(1 − 0.01)x
5. y = 2.05(0.6)x
Identify each situation as growth or decay, and evaluate the result.
6. A colony of bacteria is created with 200 bacteria, and the population doubles every hour. Find the population 1 day(24 h) later.
7. A patient is given an injection of 250 mg of a drug. Each hour, as the body metabolizes the drug, the level in the bloodstream is reduced by 20%. What is the level in the bloodstream 4 h later?
8. A city had a population of 250,000 in 2008, and the population was increasing by 11% per year. What would the population be in 2012?
9. A new car is purchased for $24,000. The car depreciates(loses value) at 12% per year. How much is the car worth 3 years later?
10. A county had 45,000 acres of forested land in 1996, but that acreage was decreasing at 5% per year. How many acres of forested land remained in the county in 2000?
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Graphing the exponential functions
Graphs of exponential functions have a characteristic shape, almost flat on one end and very steep on the other. The flat end approaches a horizontal asymptote, a horizontal line that the graph comes very close to but doesn’t touch. The graph of an exponential growth rises