Exponential and Logarithmic Functions

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Problems 3

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Topics Problems 3

Problem : Which investment earns more interest: $10,000 at 6.2% interest compounded quarterly (4 times per year) for twenty years, or $10,000 at 7.5% interest compounded continuously for 15 years?

$10,000 at 6.2% interest compounded quarterly (4 times per year) for twenty years: A(t) = 10, 000(1 +

)⁸⁰

34, 229 dollars. $10,000 at 7.5% interest compounded continuously for 15 years: A(t) = 10, 000e^1.125

30, 802. The first investment earns more interest.

Problem : How soon will an investment double its value if it is invested at 5% compounded monthly?

In about 13.89 years the value of the investment will have doubled.

Problem : A city's population grows at the constant relative rate of 6%. If the original population is 10, 000, how many people are there after ten years?

P(t) = 10, 000e^.06t.10, 000e^.60

18, 221 people.

Problem : The population of a bacteria culture exponentially grows from 2,000 to 20,000 in six hours. What is the constant relative rate of growth?

P(t) = P(0)e^kt.20, 000 = 2, 000e^6k.e^6k = 10.k =

.3838. The growth rate is approximately 38.38%.

Problem : A substance decays exponentially and has a half-life of 2200 years. A specimen is found containing only 20% of its original amount of the substance. How old is the specimen?

M(2200) = M(0)e^2200k =