Math 152, Intermediate Algebra Practice Problems #13 Instructions: These problems are intended to give you practice with the types Joseph Krause and level of problems that I expect you to be able to do. Work these problems carefully Cabrillo College, Office 711E and completely. Seek assistance in the case of difficulty. These problems will not be Phone: 479 5062 collected or graded ... they are for your benefit only. Email: jokrause@cabrillo.edu 1) The following table lists several countries, their 2002 population, and their annual percentage growth rates. Use this information (and the P = P0 (1+ r)

t template) to predict the population of each country in the year 2015. Source: Statistical Abstract of the United States

Country 2002 Population Growth Rate Predicted Population in 2015

Australia 19.5 million 0.9%

Chad 8.9 million 2.9%

Denmark 5.4 million 0.3%

India 1034.2 million 1.4%

Liberia 3.3 million 2.2%

Mexico 102.5 million 1.2%

Netherlands 16.1 million 0.5%

Saudi Arabia 23.5 million 3.3%

Spain 40.2 million 0.1%

2) The following function predicts the value, V, of a particular car after t years of ownership ... V (t) = 28400(0.82)t a) What was the value of the car when it was purchased? b) Predict the value of the car after 5 years of ownership. c) On average, how much did the value of the car drop each year over the first 10 years of ownership? Hint: Don’t over-think this and do more work than necessary … it’s actually pretty easy to answer this question. 3) Phosphorus (P32) is a radioactive isotope with a half-life of about 14 days. Suppose 40 mg of this substance is initially present. The amount that remains radioactive after t days is approximated by A(t) = 40e−0.05 t . Use this function to complete the following table.

t 0 1 2 7 14 30 365

A(t)

4) Suppose $1000 is invested in an account that earns 6.4% interest compounded monthly. The balance, B, of the account after t years is given by B(t) = 1000(1.0053)12 t . Use this function to complete the following table.

t 0 5 10 15 20 25 30

B(t)

5) Suppose $1000 is invested in an account that earns 6.4% interest compounded continuously. The balance, B, of the account after t years is given by B(t) = 1000e0.064 t . Use this function to complete the following table.

t 0 5 10 15 20 25 30

B(t)

6) The function T (m) = 70 +110e −0.032m , which is based on Newton’s Law of Cooling, models how a baked chicken cools after being removed from an oven. In this model, T is the temperature in degrees Fahrenheit and m is the number of minutes it has been out of the oven. Use this function to complete the following table.

t 0 5 10 15 20 25 30

T(m)

7) Solve each of the following exponential equations by explicitly showing that the bases are the same.

a) 5x = 25 b) 3x = 81 c) 2x+4 = 8 d) 23x = 14

e) 3−x = 13

f) 2x2 −3x = 16 g) 109x = 1000 h) 101−2x = 0.01

i) 3211789

⎛⎝⎜

⎞⎠⎟x= 1

Math 152, Intermediate Algebra Practice Problems #14 Instructions: These problems are intended to give you practice with the types Joseph Krause and level of problems that I expect you to be able to do. Work these problems carefully Cabrillo College, Office 711E and completely. Seek assistance in the case of difficulty. These problems will not be Phone: 479 5062 collected or graded ... they are for your benefit only. Email: jokrause@cabrillo.edu 1) Find the value of each logarithm.

a) log218

⎛⎝⎜

⎞⎠⎟ b) log4 16( ) c) log3(81) d) log5

1125

⎛⎝⎜

⎞⎠⎟

e) log6 (216) f) log2 (64) g) log(10) h) log(0.001) i) log(1) j) ln(e) k) ln(1) l) ln(eπ ) 2) Use your calculator to approximate each logarithm. Round your answers to 4 decimal places. a) log(40) b) ln(40) c) log(0.5) d) log(95)

e) ln(6) f) ln(2.72) g) log 13

⎛⎝⎜

⎞⎠⎟ h) log(0)

i) log5 (10) j) log2 (26) k) log3.4 (2) l) log0.8 (32)

m) 3+ log8 (22) n) 35.5 + log(12)2.5

o) 5 − ln(125)1.5

p) 72 − log3(12.8)6

3) Rewrite each of the following logarithms in exponential form. a) log2 (1024) = 10 b) log3(6) ≈ 1.63 c) log2 (5) ≈ 2.32 d) ln(3) ≈ 1.1

e) log(100) = 2 f) log(90) ≈ 1.95 g) log5125

⎛⎝⎜

⎞⎠⎟ = −2 h) ln(1) = 0

4) Rewrite each of the following in logarithmic form. a) 25 = 32 b) 62 = 36 c)10−2 = 0.01 d) 82.5 ≈ 181 e) e3 ≈ 20 f) 31.2 ≈ 3.74 g) e0 = 1 h) b1 = b

Math 152, Intermediate Algebra Practice Problems #15 Instructions: These problems are intended to give you practice with the types Joseph Krause and level of problems that I expect you to be able to do. Work these problems carefully Cabrillo College, Office 711E and completely. Seek assistance in the case of difficulty. These problems will not be Phone: 479 5062 collected or graded ... they are for your benefit only. Email: jokrause@cabrillo.edu 1) Solve each of the following logarithmic equations. a) log3 x = 2 b) log(x − 6) = 2 c) 3log2 (2x) = 9 d) log4 (x +1) − 2 = 0 e) log5 (x

2 + 4x) = 1 f) log2 x − 5( ) = 3 g) log3(x) + log3(x − 8) = 2 h) log2 (x) − log2 (x + 6) = 4 i) log(x) + log(20 − x) = 2 2) Solve each of the following exponential equations. Where necessary, round your answers to 4 decimal places.

a) 3x = 10 b) 42x = 20 c) ex2 = 14

d) 2x+7 − 9 = 3 e) 51− x = 24 f) 12ex−8 = 1

2e

g) 2.2x+7 − 3 = 8 h) 2.6 ⋅ 52x = 224 i) 12.5 ⋅ 3.11.02x = 20

j) 1.5x − 73

−10 = 0 k) 600e1.6k = 2000 l) 16 − e0.14t = 2.8

3) The following table shows the 2002 population (in millions) for several states and their 2002 percentage growth rates. Use this information to compute the population doubling time for each state.

State 2002 Pop 2002 Growth Rate Doubling Time

Arizona 5.5 3.2%

California 35.1 1.9%

Colorado 4.5 2.4%

Mississippi 2.9 0.5%

Nevada 2.2 4.4%

North Dakota 0.6 -0.7%

4) Suppose a radioactive substance decays according to the following model: A(t) = 158e −0.0025t . In this model, A is the amount of the substance in grams and t is time in years. a) How much of the substance was initially present? b) How much of the substance is present after 10 years? c) Find the half-life of this substance. That is, how long will it take until only half of the initial amount remains radioactive? 5) If interest is compounded “continuously”, the template A = Pert can be used to predict the future amount. Suppose $5,000 is invested in an account where the interest is compounded continuously. If $12,000 is desired after 35 years, what interest rate would be required? 6) Suppose a cup of coffee left in a room cools according to the following model:T (t) = 65 +115e −0.05t . Here, T is the temperature of the coffee in degrees Fahrenheit and t is time measured in minutes. a) What was the temperature of the coffee when it was initially placed in the room? b) What is the temperature of the coffee after 5 minutes? c) How long before the coffee tastes “cold”? Hint: It will taste cold when it drops below normal body temperature … what’s normal body temperature? Don’t know? Now what? 7) The following model is an unconstrained, exponential growth model for the population of the United States based on recent data: P(T ) = 310e 0.01T . In this model, T is the number of years after 2007 and P is the population in millions. a) According to this model, when will the U.S. population reach 400 million? b) Based on this model, when will the U.S. population be double what it was in the year 2007? Do you think this is a reasonable prediction? Why or why not? 8) Given f (x) = x + 2 , g(x) = 2x −1 , and h(x) = x2 − 4 , find… a) ( f + g)(x) b) ( f − g)(x) c) ( fh)(x) d) (gg)(x)

e) fg

⎛⎝⎜

⎞⎠⎟(x) f)

hf

⎛⎝⎜

⎞⎠⎟(x) g) ( f f )(x) h) (g h)(x) i)

i) (h g)(x) j) (h f )(x) k) (h f h)(x) l) ( f g h)(x) 9) For each pair of functions find ( f g)(x) and (g f )(x) . a) f (x) = 2x − 3, g(x) = x + 4 b) f (x) = 1

3 x − 5, g(x) = 3x +15 c) f (x) = x2 + 2, g(x) = 3x +1

d) f (x) = x −14, g(x) = 4x +1 e) f (x) = x2 +1, g(x) = x −1 f) f (x) = 1

x − 3, g(x) = x2 + x

10) Given f (x) , find f −1(x) . Verify. State any and all domain restrictions. a) f (x) = 2x − 3 b) f (x) = 1

3 x − 5 c) f (x) = 3x +1

d) f (x) = x −14

e) f (x) = x −1 f) f (x) = 1x − 3