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Math 150 – Lynch
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Week-in-Review
Sections 6A–7B
11/6/2016
1. Solve the following equations.
(a) 84x−2 = 165x+4
(b) 15 = 5
2 4x
5
(c) 8 − log3 (x + 8) = 4
(d) 5ex
2
−7
=9
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Math 150 – Lynch
Week-in-Review: 6A–7B
(e) ln(x + 3) − ln(x − 4) = 9
(f) log3 (2x − 5) = 3 + log3 (5x + 9)
(g) 4 ln x = ln 3 + ln(2x − 3) + ln(2x + 3)
(h) 12e6x log3 (2 − x) + 11e3x log3 (2 − x) − 15 log3 (2 − x) = 0
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Math 150 – Lynch
Week-in-Review: 6A–7B
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2. Suppose that a population of bacteria has an initial population of P0 and doubles
its population every 9 hours. What is the exponential growth model where t is
the time in hours?
3. A population of bacteria starts with 500 bacteria and grows to 4000 bacteria in 3
hours.
(a) Assuming the bacteria grows according to an exponential model, find a function P (t) that gives the number of bacteria in the culture after t hours.
(b) How long will it take the population to reach 1,000,000 bacteria?
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Math 150 – Lynch
Week-in-Review: 6A–7B
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4. Suppose a radioactive material has a half-life of 100 years. How many years will
it take 50 grams to decay to:
(a) 12.5 grams
(b) 8 grams
5. What is the half-life of a sample that decayed 37% after 7 years?
6. A cup of hot coffee is left on a table, and it’s temperature (in ◦ F) after t minutes
is given by T (t) = 70 + 50ekt . After 20 minutes the temperature of the coffee has
cooled to 90◦ F.
(a) Find the value of k.
(b) How long will it take for the object to cool to 75◦ F.
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Math 150 – Lynch
Week-in-Review: 6A–7B
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7. If someone invests money at 5% interest compounded continuously, how long will
it take for the investment to double?
8. At what rate r would someone need to invest their money, if they wanted the
money to double every 10 years? Assume the interest will be compounded continuously.
9. Find the point where the lines 5x − 3y = 2 and −2x + 4y = 7 intersect, if they
do.
10. Solve the following system of equations:
7x + 8y = −12
−21x − 24y = 36
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Math 150 – Lynch
Week-in-Review: 6A–7B
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11. Suppose it takes a plane four hours to fly 1600 miles from one city to another, and
it is flying into the wind. It only takes 3.5 hours to fly the return trip over the
same distance with the wind. If the plane’s speed and the speed of the wind are
constant, then what is the plane’s speed and the speed of the wind? (By plane’s
speed, we mean the speed of the plane if it was flying with no wind.)
12. A chemist has two different saline solutions. Solution 1 is 27% salt and solution
2 is 10% salt. The chemist would like to have 5 liters of 20% salt solution. How
much of each solution should the chemist mix together to get the desired solution?
13. Solve the following systems of equations.
(a) x2 + y 2 = 16
y = x2 − 4
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Math 150 – Lynch
Week-in-Review: 6A–7B
(b) 2x2 + y 2 + 5y = 0
3x2 + 2y 2 + 3y = 5
(c) (x − 2)2 − (y − 3)2 = −6
x2 + (y − 3)2 = 16
(d) 3y + x = 4
x2 − y 2 − 4y = 136
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