Question 207881This question is from textbook College Algebra and Trigonometry
: How do you solve a logarithmic word problem of this nature.
LAW A lawyer has determined that the number of people P(t)in a city of 1,200,000 people who have been exposed to a news item after t days is given by the function P(t)=1,200,000(1-e^-0.03t)
a. How many days after major crime has been reported have 40% of the population heard of the crime?
b. A defense lawyer knows it will be very difficult to pick an unbiased jury after 80% of the population have heard of the crime. After how many days will 80% of the population have heard of the crime?
This question is from textbook College Algebra and Trigonometry
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! A lawyer has determined that the number of people P(t)in a city of 1,200,000 people who have been exposed to a news item after t days is given by the function P(t)=1,200,000(1-e^-0.03t)
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PROBLEM A
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a. How many days after major crime has been reported have 40% of the population heard of the crime?
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p(t) = 1,200,000 * (1-e^(-.03*t)
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when 40% of the population hear of the crime, this means that p(t) = 480,000
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formula becomes:
480,000 = 1,200,000 * (1-e^(-.03*t)
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remove the parentheses to get:
480,000 = 1,200,000 - (1,200,000) * (e^(-.03*t))
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subtract 1,200,000 from both sides of the equation to get:
-720,000 = -(1,200,000) * (e^(-.03*t))
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divide both sides of the equation by (-1,200,000) to get:
(-720,000)/(-1,200,000) = e^(-.03*t))
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this becomes:
.6 = e^(-.03*t))
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the equation is now in a form that you can solve using logarithms.
you can use LOG function of your calculator or LN function of your calculator.
it doesn't matter which.
you'll get the same either way.
we'll use the LOG function of the calculator.
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take the log of both sides of this equation to get:
log(.6) = log(e^(-.03*t))
by the laws of logarithms, this becomes:
log(.6) = (-.03*t) * (log(e))
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divide both sides of this equation by log(e) to get:
log(.6)/log(e) = -.03*t
this becomes:
-.03*t = -.510825624
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since our exponent for e is -.03*t we'll leave the answer as is for now and just use that answer to verify that we have the right number for the exponent of e.
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the equation we are trying to solve is:
480,000 = 1,200,000 * (1-e^(-.03*t)
substituting -.510825624 for -.03*t, that equation becomes:
480,000 = 1,200,000 * (1 - e^(-.510825624))
using the calculator to find e^(-.510825624) our equation becomes:
480,000 = 1,200,000 * (1 - .6) which becomes:
480,000 = 1,200,000 * (.4) which becomes:
480,000 = 480,000 proving that the exponent we calculated was good.
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we still haven't answered the question though.
that question was how many days which is represented by t.
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since we know that -.03*t = -.510825624 then we can solve for t by getting:
t = (-.510825624) / (-.03) = 17.02752079
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our answer is that 40% of the population will have read about the crime in 17.02752079 days which rounds down to 17 days.
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PROBLEM B
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b. A defense lawyer knows it will be very difficult to pick an unbiased jury after 80% of the population have heard of the crime. After how many days will 80% of the population have heard of the crime?
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problem b can be solved using the same method as in problem a.
only the numbers will change.
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p(t) = 1,200,000 * (1-e^(-.03*t)
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when 80% of the population hear of the crime, this means that p(t) = 600,000
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formula becomes:
960,000 = 1,200,000 * (1-e^(-.03*t)
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remove the parentheses to get:
960,000 = 1,200,000 - (1,200,000) * (e^(-.03*t))
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subtract 1,200,000 from both sides of the equation to get:
-240,000 = -(1,200,000) * (e^(-.03*t))
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divide both sides of the equation by (-1,200,000) to get:
(-720,000)/(-1,200,000) = e^(-.03*t))
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this becomes:
.2 = e^(-.03*t))
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the equation is now in a form that you can solve using logarithms.
you can use LOG function of your calculator or the LN function of your calculator.
it doesn't matter which.
you'll get the same either way.
we'll use the LN function of the calculator this time, just to be different.
LN means natural logarithms which are taken to the base e.
LOG means common logarithms which are taken to the base 10.
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take the natural log of both sides of this equation to get:
ln(.2) = ln(e^(-.03*t))
by the laws of logarithms, this becomes:
ln(.2) = (-.03*t) * (ln(e))
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divide both sides of this equation by ln(e) to get:
ln(.2)/ln(e) = -.03*t
this becomes:
-.03*t = -1.609437912
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since our exponent for e is -.03*t we'll leave the answer as is for now and just use that answer to verify that we have the right number for the exponent of e.
-----
the equation we are trying to solve is:
960,000 = 1,200,000 * (1-e^(-.03*t)
substituting -1.609437912 for -.03*t, that equation becomes:
960,000 = 1,200,000 * (1 - e^(-1.609437912))
using the calculator to find e^(-1.609437912) our equation becomes:
960,000 = 1,200,000 * (1 - .2) which becomes:
960,000 = 1,200,000 * (.8) which becomes:
960,000 = 960,000 proving that the exponent we calculated was good.
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we still haven't answered the question though.
that question was how many days which is represented by t.
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since we know that -.03*t = -1.609437912 then we can solve for t by getting:
t = (-1.609437912) / (-.03) = 53.64793041
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our answer is that 80% of the population will have read about the crime in 53.64793041 days which rounds up to 54 days.
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