Applied Algebra FX01 Exam (C957)
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Free Applied Algebra FX01 Exam (C957) Questions
The population of the United States, in millions of people, can be estimated with the function P(A) = 282(1.01)(t), where t = 0 represents the population, P, in the year 2000 and t is measured in years. What is the average rate of change of the population from t = 2 to t =13?
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The population increases at a rate of 2.560 million people per year.
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The population increases at a rate of 3.025 million people per year.
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The population increases at a rate of 287.668 million people per year.
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The population increases at a rate of 320.942 million people per year.
Explanation
To find the average rate of change of the population from t = 2 to t = 13, we use the formula for the average rate of change:
Average rate of change = P(t2)−P(t1)t2−t1
Step 1: Define the function and compute P(t2) and P(t1).
The population function is given by: P(A) = 282(1.01)(t)
For t1 = 2 and t2 = 13:
Compute P(2):
P(A) = 282(1.01)(t)
P(2) = 282(1.01)(2)
= 287.668 million
Compute P(13):
P(13) = 282(1.01)(13) = 320.942 million
Step 2: Compute the average rate of change
= 3.025 million people per year
Correct Answer:
The population increases at a rate of 3.025 million people per year.
Why the other options are incorrect:
“The population increases at a rate of 2.560 million people per year”: This rate is too low and does not match the calculated value of 3.025.
“The population increases at a rate of 287.668 million people per year”: This is a misinterpretation of P(2), which is the population at t=2, not the rate of change.
“The population increases at a rate of 320.942 million people per year”: This is the population at t = 13, not the rate of change.
A small pond contains 12 units of dissolved oxygen in a fixed volume of water. At time t = 0, a quantity of organic waste is introduced into the pond. The oxygen concentration, y, is shown in t weeks in the graph below.
As time passes, what will be the ultimate concentration of oxygen in the pond?
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The oxygen concentration stabilizes near 5 units.
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The oxygen concentration stabilizes near 12 units.
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The oxygen concentration stabilizes near 24 units.
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The oxygen concentration stabilizes near 42 units.
Explanation
Correct answer:
The oxygen concentration stabilizes near 12 units.
Explanation
This is because the oxygen concentration graph flattens between 11 and 12 but seems to stop the increment at 12 units.
Why other options are wrong:
“The oxygen concentration stabilizes near 5 units” This is incorrect since the horizontal asymptote is between 11 and 12 units.
“The oxygen concentration stabilizes near 24 units” This is incorrect since the oxygen level does not reach this level.
“The oxygen concentration stabilizes near 42 units” This is incorrect since the oxygen level does not reach this level.
The graph below shows the three-month average percentage year-on-year growth for real total pay for the years 2010-2016, along with the Office for Budget Responsibility's (OBR's) growth forecast for the years 2016- 2021.
What can be concluded from this graph?
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The real total pay growth outturn exceeded 3% at some point between 2010 and 2016.
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The OBR forecasts the growth will remain between 1% and 2% between 2016 and 2021.
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The OBR forecasts the growth will remain above 1% between 2016 and 2021.
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The real total pay growth outturn remained between -2% and 2% from 2010 and 2016.
Explanation
Correct answer:
The OBR forecasts the growth will remain above 1% between 2016 and 2021.
Explanation:
This is correct since the extrapolated line does not go below 1% between 2016 and 2021.
Why other options are wrong:
“The real total pay growth outturn exceeded 3% at some point between 2010 and 2016.” This is not true since the graph does not exceed 3% on the y axis.
“The OBR forecasts the growth will remain above 1% between 2016 and 2021.” This is not true since there is no certainty but just a prediction.
“The real total pay growth outturn remained between -2% and 2% from 2010 and 2016.” This is not true as it can be seen in the graph that the growth in 2012 was below -2% and above 2% between 2015 and 2016.
For the years since a department store has been open, its annual revenue, R, (in millions of dollars) can be modeled by the function graphed below, where t is the number of years since the store opened.
How should the maximum revenue be interpreted?
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About 6 years after the department store opened, the store earned a maximum revenue of about $1,400,000.
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About 6 years after the department store opened, the store earned a maximum revenue of about $1,600,000.
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About 4 years after the department store opened, the store earned a maximum revenue of about $1,600,000.
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About 14 years after the department store opened, the store earned a maximum revenue of about $1,400,000.
Explanation
Solution:
At around 6 years after the department store was opened, the store had the highest annual revenue of $1,600,000.
Correct answer:
About 6 years after the department store opened, the store earned a maximum revenue of about $1,600,000.
Why others are wrong:
“About 6 years after the department store opened, the store earned a maximum revenue of about $1,400,000.” About 6 years after the department store opened the revenue was about $1,600,000.
“About 4 years after the department store opened, the store earned a maximum revenue of about $1,600,000.” About 4 years after the department store opened the revenue was slightly above 1,500,000.
“About 14 years after the department store opened, the store earned a maximum revenue of about $1,400,000.” About 14 years after the department store opened the revenue was zero.
A new company has tracked employee growth throughout its first seven years in operation. The company performed a logistic regression and modeled the number of employees, N, working at the company t years after the company started with the function and with r2 = 0.97.
Would it be appropriate for the company to use this model to make a prediction about the number of employees that will be working at the company 14 years after it started?
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No, it would not be appropriate because this data should be modeled by a quadratic function.
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Yes, it would be appropriate because r2 is close to 1.
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Yes, it would be appropriate because employee growth should be modeled by a logistic function.
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No, it would not be appropriate because t = 14 is too far outside the range of known values; this would be extreme extrapolation.
Explanation
Correct Answer:
No, it would not be appropriate because t = 14 is too far outside the range of known values; this would be extreme extrapolation.
Explanation:
Extrapolation involves making predictions about values outside the range of the observed data by assuming that the existing trend or pattern will continue. This is problematic because, in many cases, real-world data does not follow a constant or predictable pattern indefinitely. External factors or changes in underlying conditions can cause the trend to shift or behave unpredictably, leading to inaccurate predictions. Thus, predicting too far beyond the last data point without considering possible changes in the trend can result in unreliable forecasts.
Why other options are wrong:
“No, it would not be appropriate because this data should be modeled by a quadratic function.” This is incorrect because the logistic function is the appropriate model for this type of data. Employee growth in a company often follows an S-shaped curve, which logistic regression captures effectively. A quadratic function does not align with the observed pattern or behavior of growth data.
“Yes, it would be appropriate because r2 is close to 1.” This is incorrect because while r2 = 0.97 indicates a strong fit within the observed data range (up to t = 7), making predictions for t = 14 involves extreme extrapolation beyond the range of known data, which reduces reliability regardless of the r2 value.
“Yes, it would be appropriate because employee growth should be modeled by a logistic function.” This is incorrect because, while a logistic function is suitable for modeling employee growth, the appropriateness of predictions depends on whether the predictions fall within or near the range of observed data. Since t = 14 is well beyond the observed range (t = 1 to t = 7), predictions become unreliable.
Two companies, Ataron and Endothon, produce carbon fibers. For both companies, the price to produce a carbon fiber depends on its length, t. The exponential function that models Alaron's price, A(t), is shown with the solid curve below. The exponential function that models Endothon's price, N(t), is given with the dashed line below.
Based on this graph, which company has a lower production price for a carbon fiber that is 5 inches long?
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Ataron has a lower production cost, at about $190 compared to Endothon's price of about $200.
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Endothon has a lower production cost, at about $190 compared to Ataron's price of about $200.
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Ataron has a lower production cost, at about $116 compared to Endothon's price of about $141.
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Endothon has a lower production cost, at about $116 compared to Ataron's price of about $141.
Explanation
Correct answer:
Endothon has a lower production cost, at about $116 compared to Ataron's price of about $141.
Explanation:
From the graph the company that is represented by the model with dashed line has the lowest price for 5 inches at about slightly less than $120 compared by Ataron’s price of slightly above $140.
Why other options are wrong:
“Ataron has a lower production cost, at about $190 compared to Endothon's price of about $200” This is incorrect since the range of the y axis in the graph is from 0 to 160 and 1 to 10 on the x axis.
“Endothon has a lower production cost, at about $190 compared to Ataron's price of about $200” This is incorrect since the range of the y axis in the graph is from 0 to 160 and 1 to 10 on the x axis.
“Ataron has a lower production cost, at about $116 compared to Endothon's price of about $141” This option interchange the companies hence incorrect.
Company A wanted to compare its quarterly revenue against its competitor's revenue. The table below shows the revenue for both companies.
What can be concluded about the revenues of the two companies?
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Company A's higher revenues occurred in the second half of the year. Company B's higher revenues occurred in the first half of the year.
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Both Company A and Company B had higher revenues in the second half of the year than in the first half of the year.
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Both Company A and Company B had higher revenues in the first half of the year than in the second half of the year.
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Company A's higher revenues occurred in the first half of the year. Company B's higher revenues occurred in the second half of the year.
Explanation
Correct Answer:
Company A's higher revenues occurred in the first half of the year. Company B's higher revenues occurred in the second half of the year.
Explanation:
Company A Revenue:
First half (Q1 + Q2): 80,000 + 110,000 = 190,000
Second half (Q3 + Q4): 70,000 + 67,000 = 137,000
Conclusion: Higher revenues occurred in the first half.
Company B Revenue:
First half (Q1 + Q2): 64,000 + 97,000 = 161,000
Second half (Q3 + Q4): 96,000 + 95,000 = 191,000
Conclusion: Higher revenues occurred in the second half.
Why the Other Options Are Wrong:
"Company A's higher revenues occurred in the second half of the year. Company B's higher revenues occurred in the first half of the year." This is incorrect because Company A's higher revenues were in the first half, not the second, and Company B's higher revenues were in the second half, not the first.
“Both Company A and Company B had higher revenues in the second half of the year than in the first half of the year.” This is incorrect because only Company B had higher revenues in the second half. Company A had higher revenues in the first half.
"Both Company A and Company B had higher revenues in the first half of the year than in the second half of the year." This is incorrect because only Company A had higher revenues in the first half. Company B had higher revenues in the second half.
The annual unemployment rate, U(t), of a certain U.S. city from 1995 to 2015 is shown on the graph below, with t = 0 corresponding to 1995.
What is the average rate of change from t = 6 to t = 12?
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There is on average a 0.833% increase in unemployment every year.
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There is on average a 0.167% increase in unemployment every year.
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There is on average a 0.666% increase in unemployment every year.
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There is on average a 1% increase in unemployment every year.
Explanation
Correct answer:
There is on average a 0.167% increase in unemployment every year.
Solution:
= 0.1666
Why other are wrong:
“There is on average a 0.833% increase in unemployment every year.” This is incorrect because the average rate of change is only 0.167%, not 0.833%. This value is too large and doesn't match the calculated value.
“There is on average a 0.666% increase in unemployment every year.” This is also incorrect because 0.666% is larger than the calculated 0.167% rate. It suggests a much greater increase than the one observed.
“There is on average a 1% increase in unemployment every year.” This option is also incorrect because the calculated rate of change is much lower than 1%. It would imply a significantly higher increase than what is actually observed between 2001 and 2007.
A mathematical model was used to predict the spread of a bacterial infection, B, in a population based on temperature (measured in t degrees Celsius). The function was B(t) = 250 + 45e-0.2t
According to this model, are there more infections at 15°C or 35°C?
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More infections at 15°C since B(15) > B(35)
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More infections at 35°C since B(15) > B(35)
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More infections at 15°C since B(15) < B(35)
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More infections at 35°C since B(15) < B(35)
Explanation
Solution:
To determine whether there are more infections at 15°C or 35°C, we need to compare B(15) and B(35) using the given function:
B(t) = 250 + 45e-0.2t
Step 1: Compute B(15) Substitute t = 15 into the function:
B(15) = 250 + 45e-(0.2 × 15)
≈ 250 + 45(0.0498)
≈ 252.24
Step 2: Compute B(35) Substitute t = 35 into the function:
B(35) = 250 + 45e-(0.2 × 35)
≈ 250 + 45(0.0009)
≈ 250.04
Step 3: Compare B(15) and B(35)
B(15) ≈ 252.24 B(35)
≈ 250.04
Since B(15) > B(35), there are more infections at 15°C than at 35°C.
Correct Answer:
More infections at 15°C since B(15) > B(35)
Why the Other Options Are Incorrect:
"More infections at 35°C since B(15) > B(35)": This contradicts the conclusion. If B(15) > B(35), there are more infections at 15°C, not 35°C.
"More infections at 15°C since B(15) < B(35)": The inequality is incorrect. B(15) is greater than, not less than, B(35).
"More infections at 35°C since B(15) < B(35)": Both the conclusion and inequality are incorrect. B(15) > B(35), and this means more infections at 15°C.
The four lines on the graph below represent the total value, y, of purchased equipment x years after purchase for four different pieces of equipment.
Which piece of equipment loses its value the most slowly?
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D
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A
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B
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C
Explanation
Correct option: D
Explanation:
This is the only line with the highest rate of change in x with respect to y as compared to the others.
Why other options are wrong:
“A” Line A seems to be having a higher gradient than line D since it slants more, hence this option is not correct.
“B” Line B seems to be having a higher gradient than line D since it slants more, hence this option is not correct.
“C” Line C seems to be having a higher gradient than line D since it slants more, hence this option is not correct.
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