Expert: Sherman D. - 4/22/2006

Question
1)  Use the compound interest models
A=P(1+r/n)^nt and A=Pe^rt to answer this question.  Suppose you want to invest $5000.  What investment yields the greater return over 5 years:  4.5% compounded quarterly or 4% compounded continuously?  How much more, to the nearest dollar, is yielded by the better investment?

2)  Find the domain, vertical asymptote, and the x-interept of f(x)=log4(5-x)

3)  Evalute each of the following problems.

a)  log4 32

b) log8 (1/16)

c) 2^log2 (5)

d)  logb (1/b^3)

4.  Strontium-90 is a waste product from nuclear fission reactors.  It's half life is 28 years and its decay model is A=Aoe^-0.024755t.  Suppose a nuclear accident occurs and releases 50 grams of strontium-90 into the biosphere.  How long will it take for strontium-90 to decay to a level of 20 grams?

5.  If there is no restriction on food and living space, the rate of growth of a population of living organisms is proportional to the size of the population.  Thus, the formula A=Aoe^kt is appropriate.  Assume that in the absence of hunters, the deer population in New York State triples every 11 years.  Find the size of the herd of 500 deer after 5 years.


Answer
1) Use the compound interest models
A=P(1+r/n)^nt and A=Pe^rt to answer this question. Suppose you want to invest $5000. What investment yields the greater return over 5 years: 4.5% compounded quarterly or 4% compounded continuously? How much more, to the nearest dollar, is yielded by the better investment?

A = 5000(1 + (.045/4))^(5 * 4)
A = 5000(1 + .01125)^20
A = 5000(1.01125)^20
A = about $6253.75

A = 5000e^(.04 * 5)
A = 5000e^(.2)
A = about $6107.01

5000 at 4.5% compounded quarterly for 5 years is the better deal. Yielding an extra $146.74.

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2) Find the domain, vertical asymptote, and the x-interept of f(x)=log4(5-x)

f(x) = log(4)(5 - x)
f(x) = log(4)(-x + 5)
same as saying
f(x) = (log(-x + 5))/(log(4))

Using www.calculator.com/calcs/GCalc.html

Domain : x < 5
VA : x = 5
x-intercept : x = 4

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3) Evalute each of the following problems.

a) log(4)32 = (log(32))/(log(4)) = 2.5

b) log(8)(1/16) = (log(1/16))/(log(8)) =  (-4/3) or -1.3333

c) 2^(log(2)5) = 2^((log(5))/(log(2))) = 5

d) logb (1/b^3) = log(b)(b^-3) = -3log(b)(b) = -3((log(b))/(log(b))) = -3(1) = -3

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4. Strontium-90 is a waste product from nuclear fission reactors. It's half life is 28 years and its decay model is A=Aoe^-0.024755t. Suppose a nuclear accident occurs and releases 50 grams of strontium-90 into the biosphere. How long will it take for strontium-90 to decay to a level of 20 grams?

20 = 50e^(-.024755t)
(2/5) = e^(-.024755t)
ln(2/5) = -.024755t
t = (ln(2/5))/(-.024755)
t = about 37 years

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5. If there is no restriction on food and living space, the rate of growth of a population of living organisms is proportional to the size of the population. Thus, the formula A=Aoe^kt is appropriate. Assume that in the absence of hunters, the deer population in New York State triples every 11 years. Find the size of the herd of 500 deer after 5 years.

A = 500e^(3(5/11))
A = 500e^(15/11)
A = about 1955 deer

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