Exponential and Logarithmic Functions

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?

Solution for Problem 1 >>

$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.

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

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?

Solution for Problem 3 >>

P(t) = 10, 000e ^.06t.10, 000e ^.60 18, 221 people.

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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?

Solution for Problem 4 >>

P(t) = P(0)e ^kt.20, 000 = 2, 000e ^6k.e ^6k = 10.k = .3838 . The growth rate is approximately 38.38%.

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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?

Solution for Problem 5 >>

M(2200) = M(0)e ^2200k = M(0).k = - .000315067.M(t) = M(0)e ^-.000315067t = .2M(0).t = 5100 years. The specimen is about 5100 years old.

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