Test 2 ReviewUse complete sentences to "explain or interpret". Round final answers to 2 decimal places(unless stated otherwise). Don’t forget units. Circle or box your final answer (meaning: whenI grade your test, make sure I understand what you have decided to be the answer for theproblems).

1. Bob wants to invest $2000 in a savings account that has an interest rate of 5.5% per year, compoundedsemi-annually.

(a) Write the function that will determine the amount of money in the account after an amount oftime.

(b) How much money will be in the savings account after 28 months?(c) How many years would it take for the investment to reach $4050?

2. A certain population has a yearly per capita growth rate of 2.3%, and the initial value is 3 million.

(a) Write the function of P, the population in millions, as a function of the years t.(b) Find the population after 4 years.

3. A lake is 50% full of fish. The number of fish are increasing at a relative growth rate of 5% every year.

(a) If the fish continue on this trend, write a function that models the percentage of the fish populationin the lake.

(b) How long will it take for the lake be 80% full of fish?

4. Solve the equations.

(a) 32x−7 = 27

(b) log2(1− x) = 4

(c) log8(x+ 5)− log8(x− 2) = 1

(d) 41−x = 32x+5

(e) e3x/4 = 10

(f) log(x) + log(x+ 1) = log(12)

5. You are tight on cash and need to pay the bills quick, which have accumulated to $5000. So you call upyour local loan shark. You will lend you the $5000 at interest rate of 15% APR, compounded weekly.

(a) Write the function of A, the amount you owe the loan shark, in terms of the years, t.(b) If you haven’t paid any of your debt off, how much do you owe after 5 years?(c) How many years would it take for the debt to triple (Again, if you haven’t paid any of your debt

off)?

6. The number of students at time t (in years since 1995) at a high school with a cell phone is give bythe function

N(t) =1200

1 + 2e−0.29t

(a) In what year is the number of students with a cell phone equal to 600?(b) How many students had a cell phone initially (in the year 1995)?

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7. According to autos.aol.com, the value of a 2014 Honda Accord EX 4-door Sedan is $24,880. Accordingto Edmonds.com, the resale value of this car is modeled by the equation V (t) = 24880(0.925)t, wheret is the number of years since 2014.

(a) At what percentage rate is the car’s value decreasing every year?(b) After 3 years, what will the value of the Honda Accord EX be?(c) Assuming the trend continues, in what year will the value of the 2014 Honda Accord EX reach

$13000?

8. State the domain, range, and asymptote of the following:

(a) f(x) = 2x+1 − 4

(b) g(x) = log3(1− 2x)− 7

(c) h(x) = 3 + 4x

(d) k(x) = 5.5 + ln(5x− 7)

9. A certain culture of a bacteria initially 25 bacteria and is observed to double every 5 hours.

(a) Find the function that models the population of the bacteria.(b) Estimate the number of bacteria after one day (round your answer down).(c) After how many hours will the bacteria count reach 1 million?

10. Evaluate the expressions without using a calculator:

(a) 10log(45)

(b) log5(625)− log5(25)

(c) log2(163)

(d) e2ln(7)

(e) log8(1)

(f) log2(6)− log2(3) + log2(2)

11. The population of Tennessee has increased at an exponential rate since 1850. The population in millionscan be modeled by the function P (t) = 1.2(1.0125)t, where t is the number of years since 1850.

(a) At what percentage rate is the population of Tennessee growing?(b) What was the population of Tennessee in 1900?(c) Looking at the exponential function, how do we know that the population is in fact growing?

12. Combine the expression into a single logarithm:

(a) log(6) + 4log(2)

(b) log4(x− 2) + log4(x+ 2)− 12 log4(x

2 + 4)

(c) 3(ln(x− 4) + 5ln(x2 + 4x)

)13. Expand the logarithmic expression.

(a) ln(AB2C3)

(b) log2

(x√y2 + 1

)(c) log5

((1− 5y)3

x3y5

)

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14. The stray-cat population in a small town grows exponentially. In 1999 the town had 30 stray cats, andthe relative growth rate was 15% per year.

(a) Find a function that models the stray-cat population.

(b) Find the project population after 4 years (round your answer down).

(c) Find the number of years required for the stray-cat population to reach 500.

15. A grey squirrel population was introduced in a certain county of Great Britain 30 years ago. Biologistsobserve that the population doubles every 6 years, and now the population is 100,000.

(a) What was the initial size of the squirrel population.

(b) Write a function that models the grey squirrel population over time.

(c) Estimate the grey squirrel population 10 years from now (round your answer down).

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Test 2 Review Key1. (a) B(t) = 2000(1 + 0.055

2 )2t

(b) $2269.93(c) 13 years

2. (a) P (t) = 3(1.023)t

(b) 3.29 million

3. (a) n(t) = 0.5e0.05t

(b) 9.4 years

4. (a) x = 5

(b) x = −15

(c) x = 3

(d) x = log(4)−5log(3)2log(3)+log(4) ≈ −1.15

(e) x = 43 ln(10) ≈ 3.07

(f) x = 3

5. (a) A(t) = 5000(1 + 0.1552 )52t

(b) $10573.58(c) 7.33 years

6. (a) 2.39 years after 1995 (in the year 1997)(b) 400 students

7. (a) 7.5%(b) $19691.35(c) In the years 2022 (2022.33)

8. (a) Domain: (−∞,∞) Range: (−4,∞) Asymptote: y = −4(b) Domain: (−∞, 0.5) Range: (−∞,∞) Asymptote: x = 0.5

(c) Domain: (−∞,∞) Range: (3,∞) Asymptote: y = 3

(d) Domain: (1.4,∞) Range: (−∞,∞) Asymptote: x = 1.4

9. (a) n(t) = 25(2t/5)

(b) 696 bacteria (actual 696.44 bacteria)(c) 76.4 hours

10. (a) 45

(b) 2

(c) 12

(d) 49

(e) 0

(f) 2

11. (a) 1.25%(b) 2.23 million people(c) The factor of the exponential function, 1.0125, is greater than 1.

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12. (a) log(96)

(b) log4

((x− 2)(x+ 2)√

x2 + 4

)(c) ln

((x− 4)3(x2 + 4x)15

)13. (a) log(A) + 2log(B) + 3log(C)

(b) log2(x) +12 log2(y

2 + 1)

(c) 3log5(1− 5y)− 3log5(x)− 5log5(y)

14. (a) n(t) = 30e0.15t

(b) 54 cats (actual number is 54.66 cats)

(c) 18.76 years

15. (a) 3125 squirrels

(b) n(t) = 3125(2t/6)

(c) 317,480 grey squirrels

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