integrated math 3b

Which of the following is true of the location of the terminal side of an angle, 𝛳, whose sine value is 1/2?

𝛳 has a reference angle of 30° and is in Quadrant I or II //option A

Which expression converts 𝜋/4 radians to degrees?

𝜋/4 (180°/𝜋) //option B

Isabel deposits $6,000 into an account that earns 1.5% interest compounded monthly. Assuming no more deposits and no withdrawals are made, how much money is in the account after 4 years? Compound interest formula: V(t) = P(1 + r/n)^nt t = years since initial deposit n = number of times compounded per year r = annual interest rate (as a decimal) P = initial (principal) investment V(t) = value of investment after t years

$6,370.78 //option B

Which point corresponds to the real zero of the graph of y = log3(x + 2) - 1?

(1, 0) //option B

Tenisha solved the equation below by graphing a system of equations. log3(5x) = log5(2x + 8) Which point approximates the solution for Tenisha's system of equations?

(1.0, 1.4) //option B

The terminal side of an angle measuring 𝜋/6 radians intersects the unit circle at what point?

(√3/2, 1/2) //option A

What are two possible measures of the angle below?

-270° and 90° //option B

What is the approximate value of x in the equation below. log3/4(25) = 3x - 1

-3.396 //option A

What are two possible measures of the angle below?

-90° and 630° //option A

What is the approximate value of q in the equation below? q + log2(6) = 2q + 2

0.585 //option C

Angle T has a measure between 0° and 360° and is coterminal with a -710° angle. What is the measure of angle T?

10° //option B

What is 5𝜋/6 radians converted to degrees? If necessary, round your answer to the nearest degree.

150° //option C

Jacques deposited $1,900 into an account that earns 4% interest compounded semiannually. After t years, Jacques has $3,875.79 in the account. Assuming he made no additional deposits or withdrawals, how long was the money in the account? Compound interest formula: V(t) = P(1 + r/n)^nt t = years since initial deposit n = number of times compounded per year r = annual interest rate (as a decimal) P = initial (principal) investment V(t) = value of investment after t years

18 years //option C

Which expression can be used to determine the reference angle for an angle, x, measuring 150°?

180 - x //option A

What common base can be used to rewrite each side of the equation 2^(x + 3) - 3 = 5

2 //option A

The population of a town grew from 20,000 to 28,000. The continuous growth rate is 15%. The equation 20,000e^0.15t = 28,000 represents the situation, where t is the number of years the population has been growing. About how many years has the population of the town been growing? Use a calculator and round your answer to the nearest whole number.

2 years //option A

Given log3(2)= 0.631 and log3(7)= 1.771, what is log3(14)?

2.402 //option C

The amount of a sample remaining after t days is given by the equation P(t) = A(1/2)^t/h, where A is the initial amount of the sample and h is the half-life, in days, of the substance. A scientist has a 10-mg sample of a radioactive isotope. The isotope has a half-life of 8 days. After 16 days, how much of the radioactive isotope remains?

2.5 mg //option B

On April 11, 2012, two earthquakes were measured off the northwest coast of Sumatra. The first had a magnitude of 8.6. The second had a magnitude of 8.2. By what approximate factor was the intensity of the first earthquake greater than the intensity of the second earthquake? M = log(I/Iv0) M = the magnitude of an earthquake I = the intensity of an earthquake Iv0 = the smallest seismic activity that can be measured, which is 1

2.51 //option D

What is 4 radians converted to degrees? If necessary, round your answer to the nearest degree.

229° //option C

A type of plant is introduced into an ecosystem and quickly begins to take over. A scientist counts the number of plants after m months and develops the equation P(m) = 19.3(1.089)^m to model the situation. Most recently, the scientist counted 138 plants. Assuming there are no limiting factors to the growth of the plants, about how many months have passed since the plants were first introduced?

23.1 //option D

Over time, the number of organisms in a population increases exponentially. The table below shows the approximate number of organisms after y years. The environment in which the organism lives can support at most 600 organisms. Assuming the trend continues, after how many years will the environment no longer be able to support the population?

24 //option B

The graph shows the decibel measure for sounds depending on how many times as intense they are as the threshold of sound. Noise in a quiet room is 500 times as intense as the threshold of sound. What is the decibel measurement for the quiet room?

28 decibels //option B

Which equation is equivalent to 4^x+3 = 64?

2^2x+6 = 2^6 //option B

What is the value of cos 𝛳 in the diagram below?

3/5 //option A

What is log15(2^3) rewritten using the power property?

3log15(2) //option D

Which of the following are in the correct order from least to greatest?

3𝜋/10, 60°, 𝜋/2 2𝜋/3, 255° //option A

Carlene is saving her money to buy a $500 desk. She deposits $400 into an account with an annual interest rate of 6% compounded continuously. The equation 400e^0.06t = 500 represents the situation, where t is the number of years the money needs to remain in the account. About how long must Carlene wait to have enough money to buy the desk? Use a calculator and round your answer to the nearest whole number.

4 years //option A

The equation for the pH of a substance is pH = -log[H+], where H+ is the concentration of hydrogen ions. A basic solution has a pH of 11.2. An acidic solution has a pH of 2.4. What is the approximate difference in the concentration of hydrogen ions between the two solutions?

4.0 x 10^-3 //option B

Which expression finds the measure of an angle that is coterminal with a 45° angle?

45° + 360° //option D

Which expression converts 45° to radians?

45°(𝜋/180°) //option C

Which measure is of an angle that is coterminal with a 135° angle?

495° //option C

Which expression is equivalent to logw[(x^2-6)^4]/3sqrt(x^2+8)?

4logw(x^2-6) - 1/3logw(x^2+8) //option C

What is 720° converted to radians?

4𝜋 //option D

The graph shows the growth rate of a certain bacteria in a lab, where the number of bacteria depends on the number of hours since the start of the experiment. Based on the graph, what is the approximate number of bacteria after 16 hours?

60 bacteria //option C

What is the reference angle for a 240° angle?

60° //option B

What is 𝜋/3 radians converted to degrees? If necessary, round your answer to the nearest degree.

60° //option C

How long is the arc intersected by a central angle of 𝜋/2 radians in a circle with a radius of 4.5 cm? Round your answer to the nearest tenth. Use 3.14 for 𝜋.

7.1 cm //option D

The amount of a sample remaining after t days is given by the equation P(t) = A(1/2)^t/h, where A is the initial amount of the sample and h is the half-life, in days, of the substance. A sample contains 60% of its original amount of Fermium-257. The half-life of Fermium-257 is about 100 days. About how old is the sample?

74 days //option C

Which expression converts 8 radians to degrees?

8(180°/𝜋) //option D

The equation for the pH of a substance is pH = -log[H+], where H+ is the concentration of hydrogen ions. What is the approximate pH of a solution whose hydrogen ion concentration is 2 x 10^-9?

8.7 //option C

An angle whose measure is -302° is in standard position. In which quadrant does the terminal side of the angle fall?

Quadrant I //option A

An angle whose measure is 40° is in standard position. In which quadrant does the terminal side of the angle fall?

Quadrant I //option A

An angle whose measure is -102° is in standard position. In which quadrant does the terminal side of the angle fall?

Quadrant III //option C

Which of the following best explains why tan(5𝜋/6) ≠ tan(5𝜋/3)?

The angles do not have the same reference angle. //option A

A student solved the equation below by graphing. log6(x - 1) = log2(2x + 2) Which statement about the graph is true?

The curves do not intersect. //option A

Which of the following is true for f(x) = 5cos(x) +1?

The range of the function is the set of real numbers -4 ≤ y ≤ 6. //option D

Which of the following explains why cos60° = sin30° using the unit circle?

The side opposite a 30° angle is the same as the side adjacent to a 60° angle in a right triangle. On a unit circle, the y (sin) distance of a 30° angle is the same as the x (cos) distance of a 60° angle. //option A

The equation for the pH of a substance is pH = -log[H+], where H+ is the concentration of hydrogen ions. Which equation models the concentration of hydrogen ions (H+) of a solution whose pH is 7.2?

[H+] = 10^-7.2 //option C

What is the range of y = -5sin(x)?

all real numbers -5 ≤ y ≤ 5 //option A

What is the range of the function y = 4e^x?

all real numbers greater than 0 //option A

The pH of a particular solution is given by pH = -log(x - 2), where x represents the concentration of the hydrogen ions in the solution, in moles per liter. Which graph models the pH of this solution?

first graph //option A

When Akbar's son was four years old, Akbar put $250 into an account that guaranteed a 10% annual return. The equation f(x) = 250(1.10)^x-4 represents the amount of money in the account when the child will be x years old. Which graph models the scenario?

fourth graph //option D

What is 3 ln 3 - ln 9 expressed as a single natural logarithm?

ln 3 //option A

Which expression is equivalent to log18 - log(p + 2)?

log(18/p+2) //option B

Which expression results when the change of base formula is applied to log4(x + 2)?

log(x + 2)/log4 //option A

Which expression is equivalent to log3(x + 4)?

log3 + log(x + 4) //option C

Which of the following shows the equation log4(x + 6) = 2 rewritten using logarithms?

log4(x + 6) = log4(16) //option C

What is 2log5(5x^3) + 1/3log5(x^2 + 6) written as a single logarithm?

log5(25x^6)[3sqrt(x^2+6)] //option B

What is log5(4)(7) + log5(2) written as a single log?

log5(56) //option D

Which expression is equivalent to log8[4a(b-4/c^4)]?

log8(4) + log8(a) + (log8(b-4) - 4log8(c)) //option B

Sam is proving the product property of logarithms. Which expression and justification completes the third step of her proof?

logb(b^x+y); multiplication rule of exponents //option C

The proof for the power property of logarithms appears in the table with an expression missing. Which expression is missing from the proof?

logb(b^xr) //option C

Which is equivalent to log2(n) = 4?

logn = 4log2 //option D

An angle measuring (525n)° is in standard position. For which value of n will the terminal side fall on the y-axis?

n = 6 //option D

The estimated number of organisms in a population after t days is shown in the table below. Which equation best models the situation?

n = 600(1.2)^t //option B

Which equation gives the length of an arc, s, intersected by a central angle of 3 radians in a circle with a radius of 4 in.?

s = 4(3) //option D

A pizza is taken out of an oven and placed on a counter. The temperature, T, in degrees Fahrenheit, of the pizza after m minutes is modeled by the function T = 72 + 200e^-0.045m. Which graph represents the model?

second graph //option B

The magnitude of an earthquake, R, can be measured by the equation R = log(A/T) + D, where A is the amplitude in micrometers, T is measured in seconds, and D accounts for the weakening of the earthquake due to the distance from the epicenter. If an earthquake occurred for 4 seconds and D = 2, which graph would model the correct amount on the Richter scale?

second graph //option B

Which of the following shows the graph of y = 2 ln x?

second graph //option B

Which expression is equivalent to sin(7𝜋/6)?

sin(11𝜋/6) //option D

Food poisoning occurs when bacteria like Salmonella and Listeria grow on food. These microorganisms grow at an exponential rate. When an initial population of 100 bacteria are left at 98.6 F, they will grow at a rate of b = 100(8)^h. Which graph could be used to calculate the number of bacteria after a certain number of hours?

third graph //option C

Which angle is in standard position?

third graph //option C

Which of the following shows the graph of y = 2e^x?

third graph //option C

Which statement is true for log3(x + 1) = 2?

x + 1 = 3^2 //option A

What are the potential solutions to the equation below? 2ln(x + 3) = 0

x = -2 and x = -4 //option B

Which of the following shows the extraneous solution to the logarithmic equation log7(3x^(3) + x) - log7(x) = 2?

x = -4 //option B

What is the true solution to the logarithmic equation below? log4[log4(2x)] = 1

x = 128 //option D

What is the solution to log2(2x^3 - 8) - 2log2(x) = log2(x)?

x = 2 //option B

What is the true solution to 2 ln 4x = 2 ln 8?

x = 2 //option C

What are the potential solutions of log4(x) + log4(x + 6) = 2?

x = 2 and x = -8 //option C

What is the true solution to 2 ln e^ln(5x) = 2 ln 15?

x = 3 //option B

What is the solution to log3(x + 12) = log3(5x)?

x = 3 //option C

Which of the following shows the true solution to the logarithmic equation below? log(x) + log(x + 5) = log(6x + 12)

x = 4 //option B

What is the solution to the equation below? 3log4(x) = log4(32) + log4(2)

x = 4 //option C

Maria wrote the equation log(x/2) + log(20/x^2) = log8. What is the solution to Maria's equation?

x = 5/4 //option C

What is the true solution to the equation below? 2 ln e^ln(2x) - ln e^ln(10x) = ln 30

x = 75 //option B

What is the solution to log5(10x - 1) = log5(9x + 7)?

x = 8 //option D

What is the solution to log2(9x) - log2(3) = 3?

x = 8/3 //option B

What is the solution to 3 + 4e^x + 1 = 11

x = ln2 - 1 //option A

Which equation would have real zero(s) corresponding to the x-intercept(s) of the graph below?

y = -2^x + 4 //option A

Which equation is represented by the graph below?

y = ln x //option C

Which equation is represented by the graph below?

y = ln x + 4 //option C

Identify the equation y = ln(x) that translates five units down.

y = ln(x) - 5 //option D

Which of the following is true of the values of x and y in the diagram below?

y/x = 1 //option D

Which system of equations could be graphed to solve the equation below? log(2x + 1) = 3x - 2

y1 = log(2x + 1), y2 = 3x - 2 //option B

Consider the equation below. log4(x + 3) = log2(2 + x) Which system of equations can represent the equation?

y1 = log(x + 3)/log4, y2 = log(2 + x)/log2 //option A

Which system of equations could be graphed to solve the equation below? log0.5(x) = log3(2 + x)

y1 = logx/log0.5, y2 = log2/log3 + x //option D

Omar wants to use a graph to solve the equation below. log6(x) = log2(x + 4) Which system of equations should Omar use?

y1 = logx/log6, y2 = log(x + 4)/log2 //option D

On the unit circle, where 0 < 𝛳 ≤ 2𝜋, when is tan 𝛳 undefined?

𝛳 = 𝜋/2 and 𝛳 = 3𝜋/2 //option C