Questions and Answers
Consider circle A with radius 1 unit. If $θ=60°$, what is the length, in radians, of arc BC?
1/3
How can the collector correct his mistake in the expression for the coin's value?
The base of the simplified expression should be 1.05, and the exponent should be 2x + y.
What is the value of sin θ if cos θ = 39/89?
-80/89
What is the equation that represents the sound wave producing the note A on a piano?
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Which of the following intervals has the smallest average rate of change in the function y=2x+2?
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How many hours will it take for the number of bacteria to reach 1,000,000?
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What is the average rate of change for the interval 5-10 days for a radioactive substance?
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What is the likely purpose of the survey conducted at the rock concert?
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What values of x will result in f(x)=g(x) for the equations f(x)=2x−1 and g(x)=38x?
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What is the inverse of f(x)=−3x+12/4?
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The distribution in the number of seeds per apple is approximately normal.
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Which graph represents the equation y=−2(3)x−4?
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What is the average rate of change for the interval 10-15 days?
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Match the following sampling techniques with their descriptions:
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Study Notes
Arc Length and Circles
- For circle A with a radius of 1 unit and an angle θ of 60°, the length of arc BC is calculated as 1/3 radians.
Growth and Investments
- A rare coin purchased for $100 grows in value at 10.25% annually for x years and then at 5% for y years.
- The correct expression for the coin's value, after x+y years, is 1.1025^x * 1.05^y, where the collector originally miscalculated the exponent.
Trigonometric Values
- For an angle θ in Quadrant IV where cos θ = 39/89, the value of sin θ = -80/89 and tan θ = -80/39.
Sound Waves
- A sound wave that produces the piano note A is represented by the function y = sin(880πx), where time is on the horizontal axis.
Average Rate of Change
- The smallest average rate of change for the linear function y = 2x + 2 occurs in the interval 6 ≤ x ≤ 8.
Bacterial Growth Model
- An initial bacterial culture of 200 doubles every 33 hours, modeled as A = P(2)^(t/3). It will take approximately 37 hours to reach 1,000,000 bacteria.
Sampling Techniques
- Amanda’s simple random sample of 60 students needs to reflect the high school's social media usage accurately; various sampling methods were evaluated with yes/no answers regarding their effectiveness.
Experimentation with Fertilizers
- In a study comparing two fertilizers, Fertilizer A showed a mean radish radius of 0.52 inches smaller than Fertilizer B. A simulation indicated this difference might be due to chance, showing a probability of 0.52 for such a difference.
Exponential Growth
- Josie's text messages follow the equation t = 5 * 3^(d - 1), where t is the number of messages each day after the first.
Radioactive Substance Decay
- The average rate of change in the decay of a radioactive substance over specific intervals results in a consistent decrease per day.
Investor Survey Results
- The survey conducted revealed 82% positive impressions of the performer's song, aiming to estimate overall fan sentiments.
Logarithmic Functions
- The equation g(x) = log(x + 5) and h(x) = -x - 6 leads to a solution x = -4.9.
Intersection of Functions
- The intersection of functions f(x) = 2x - 1 and g(x) = 38x results in x-values of 0 and -2.
Function Inverses
- The inverse of f(x) = (−3x + 12)/4 is f⁻¹(x) = -4x/3 + 4.
Normal Distribution of Seeds
- Approximately 19% of apples have between 5 and 11 seeds based on a normal distribution of seed counts per apple in Hayley’s orchard.
Confidence Intervals
- The mean amount spent on oral hygiene products falls within a 95% confidence level with values including $128.30, $203.36, $206.04, and $189.85.
Graphical Representation
- The graph of the equation y = -2(3)^x − 4 can be visualized at a provided link.
Unit Circle and Angles
- On the unit circle, an angle rotation to point B corresponds to π/2 radians, indicating a quarter rotation.
Fertilizer Growth Simulation
- After testing two fertilizers, the simulation showed Fertilizer A's performance might also be influenced by the characteristics of the assigned seeds.
Exponential Decay Modeling
- Juanita’s savings account model J(t) = 500(1.015)^t has a domain representing whole numbers, as partial years are not applicable.
Comparative Growth Analysis
- The equations f(x) = 240(12)^x suggest rapid growth with particular behaviors as x approaches infinity in both directions.
Tiger Population Projections
- The equation for wild tigers in Nepal projects a population modeled by p = 121(2)^x, with future population figures requiring logarithmic transformations.
Musical Note Periodicity
- The note "A" modeled by f(x) = sin(880πx) has a period of 1/440 seconds and 880 intercepts in the interval (0, 1].
Graphical Changes and Shifts
- Analyzing the graphs of transformations shows the function 5f(x) is shifted upward compared to f(x).
Fairness of Dice Game
- Maggie's and Joe's dice game plan is biased, as results show sums greater than 6 occurring more than 50% of the time, favoring Joe.
Exponential Function Properties
- The properties of a specific exponential graph include a y-intercept at 3 and the graph's behavior indicating it is always decreasing.
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Test your knowledge with these flashcards for the Algebra 2b final exam. Covering topics from circle geometry to exponential growth, these cards provide a quick review of essential concepts. Perfect for students preparing for their finals!
