SOLUTION: Here is a doozy, my group and I have been trying to figure this one out for awhile. Exponential Decay: Water Filtration As an environmental consultant, you are responsible fo

Algebra ->  Algebra  -> Exponential-and-logarithmic-functions -> SOLUTION: Here is a doozy, my group and I have been trying to figure this one out for awhile. Exponential Decay: Water Filtration As an environmental consultant, you are responsible fo      Log On

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Question 30861: Here is a doozy, my group and I have been trying to figure this one out for awhile.
Exponential Decay: Water Filtration
As an environmental consultant, you are responsible for the remediation of effluent from a pulp mill. Each hour, a quantity of water containing 500 grams of pollutants is discharged. According to environmental regulations, the contaminent level must be reduced to less than 150 grams before the effluent can be discharged into the local river system. The effluent must pass through a series of filters to reduce the contaminant level to an acceptable value. Each time the effluent passes through one filter, 18% of the contaminants are removed.
We figured out the equation: Cn+=+500%280.82%29%5E%28n-1%29. What we can't figure out is how to re-arrange or write an equation to how many filters are required to reduce Cn to less than 150 grams. Can any one help?
My second question is one of the opposite, Exponential Growth:
Two students are saving money for a graduation trip to Asia in 3 years time. George has just sold his car for $3000 and has invested his money in a bond at an interest rate of 5.4% per year, compounded monthly. At the same time, John has invested his part-time job savings of $2700 in a GIC at an interest rate of 6.8% per year, compounded quarterly.
We indentified the equations as: George - A+=+%243000%281%2B0.054%2F12%29%5E%2812t%29, where t is the number of years of growth.
John - A+=+%242700%281%2B0.068%2F4%29%5E%284t%29, where t is the number of years of growth.
**The question we are struggling with is this: How do we re-write or write an equation to express after how many years will the two investments be approximately equal in value?
We can answer both of these by looking at our plotted points or our graphs but we were wondering if there is an Algebraic equation way of expressing them.
Thanks
Melissa
Answer by acerX(62) About Me  (Show Source):
You can put this solution on YOUR website!
Melissa the first one is solved using natural log
Using your equation Cn=500(0.82)^n-1
Cn+%3C+150
150%3C500%280.82%29%5E%28n-1%29
ln%28150%29%3Cln%28500%280.82%29%5E%28n-1%29%29 use the natural log function
ln%28150%29%3Cln%28500%29+%2B+ln%280.82%29%5E%28n-1%29 Product rule of logs
ln%28150%29%3Cln%28500%29+%2B+n-1%2Aln%280.82%29 Power rule of logs
5.01%3C6.21-.20%2A%28n-1%29 Solve for Ln
5.01%3C6.21-.20n%2B.20 distribute
5.01%3C6.41-.20n combine like terms
-1.4%3C-.20n subtract 6.41 from both sides
7%3En divide by -.20 and flip the inequality

I will look at the second question and see what I can do now.

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VIA YOUR COMMENTS I AM REWRITING THIS SOLUTION:
The second question equations are
A=3000%281%2B0.054%2F12%29%5E%2812%2At%29
A=2700%281%2B0.068%2F4%29%5E%284%2At%29
First set the two equations equal to each other:
2700%281%2B0.068%2F4%29%5E%284t%29=3000%281%2B0.054%2F12%29%5E%2812t%29 Solve what you can by combining like terms
2700%281.017%29%5E%284t%29=3000%281.0045%29%5E%2812t%29 Now introduce the natural log
ln%282700%281.017%29%5E%284t%29%29=ln%283000%281.0045%29%5E%2812t%29%29 Use Product Rule of logs
ln%282700%29%2Bln%281.017%5E%284t%29%29=ln%283000%29%2Bln%281.0045%5E%2812t%29%29 Use Power Rule of Logs
ln%282700%29%2B%284t%29%2Aln%281.017%29=ln%283000%29%2B%2812t%29%2Aln%281.0045%29 Solve the ln portions
7.90%2B%284t%29%2A0.0169=8.01%2B%2812t%29%2A0.0045 Distributive Property
7.90%2B0.0676t=8.01%2B0.054t Combine like terms
0.0136t=0.11 Solve for t
t=8.09years
Looks like they aren't going to have the same amount of money for their trip!!