Applied Algebra FX01 Exam (C957)
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Free Applied Algebra FX01 Exam (C957) Questions
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.
The function N(t) = 3t2−6t + 13 represents the number of subscribers to a streaming music service, where N is the number of subscribers (in thousands) t months after the program launched. What is the average rate of change from t=1 to t=4?
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Thirty-seven thousand customers subscribed each month.
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There were 9,000 more subscribers each month.
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Three thousand customers subscribed each month.
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Twenty-seven thousand customers subscribed each month.
Explanation
Solution:
We are tasked to find the average rate of change of the function N(t) = 3t2−6t +13 from t = 1 to t = 4.
Step 1: Recall the formula for the average rate of change
The average rate of change of a function N(t) between t = a and t = b is:
Rate of Change=N(b)−N(a) b−a
Here:
a = 1,
b = 4.
Step 2: Calculate N(1) and N(4)
Substitute t = 1 into N(t):
N(1) = 3(1)2−(6*1) + 13
= 3 - 6 + 13
= 10
Substitute t=4 into N(t):
N(4) = 3(4)2−(6*4) + 13
= 3(16) − 24 + 13
= 48−24+13
= 37
Step 3: Compute the average rate of change
Using the formula:
=
= 9
The average rate of change is 9,000 subscribers per month (since N(t) is in thousands).
Correct Answer:
There were 9,000 more subscribers each month.
Why the Other Options Are Wrong:
"Thirty-seven thousand customers subscribed each month." Incorrect because 37 is the total number of subscribers at t=4, not the average rate of change.
"Three thousand customers subscribed each month." Incorrect because the average rate of change is 9,000, note 3,000.
"Twenty-seven thousand customers subscribed each month." Incorrect because 27 is the total difference in subscribers between t=1 and t=4, not the average rate per month.
A company wants to hire more workers and determine what weekly salary will attract potential employees. The company created a model based on past hiring events, where A is a function of the weekly salary S. What can be concluded from A(1,500) = 1,000?
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If the company hires 1,500 employees, each employee will receive a weekly salary of $1,000.
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If the company hires 1,000 employees, each employee will receive a weekly salary of $1,500.
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If the company proposes a weekly salary of $1,500, then 1,000 people will apply for the position.
-
If the company proposes a weekly salary of $1,000, then 1,500 people will apply for the position.
Explanation
Correct Answer:
If the company proposes a weekly salary of $1,500, then 1,000 people will apply for the position.
Explanation:
The model A(1,500) = 1,000 indicates that when the company offers a weekly salary of $1,500, the number of applicants will be 1,000. The function A is mapping the salary (S) to the number of applicants (A), so this result shows that at a salary of $1,500, 1,000 people will apply.
Why Other Options Are Wrong:
"If the company hires 1,500 employees, each employee will receive a weekly salary of $1,000." This is incorrect because the model describes the number of applicants based on salary, not the number of employees being hired. It doesn't address the hiring process itself.
"If the company hires 1,000 employees, each employee will receive a weekly salary of $1,500." This is incorrect because the model does not specify how many employees will be hired, just the relationship between salary and applicants. The number of employees hired is not directly mentioned.
"If the company proposes a weekly salary of $1,000, then 1,500 people will apply for the position." This is incorrect because the given model, A(1,500) = 1,000, specifically relates to a salary of $1,500 and 1,000 applicants. It does not suggest a salary of $1,000 leads to 1,500 applicants.
A truck rental company determines the cost of a truck rental, C, based on the number of miles driven, m. For a customer's recent rental, the customer drove 50 miles and was charged $69.50. How would this trip be represented in function notation?
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m(50) = 69.50
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m(69.50) = 50
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C(69.50) = 50
-
C(50) = 69.50
Explanation
Correct Answer:
C(50) = 69.50
Explanation:
The cost of the rental, C, is a function of the number of miles driven, m. The notation C(50)=69.50 means that when the customer drove 50 miles, the total cost of the rental was $69.50.
Why the other options are wrong:
"m(50) = 69.50": This is incorrect because mmm represents the miles driven, not the cost function. The correct function for cost is C(m), not m(x).
"m(69.50) = 50": This is incorrect because mmm is not a function of cost. The cost C depends on the miles driven, not the other way around.
"C(69.50) = 50": This is incorrect because the function C(x) represents the cost as a function of miles driven. It does not take the cost as input.
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.
-
Both Company A and Company B had higher revenues in the first half of the year than in the second half of the year.
-
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.
A company is hosting an event in which the venue costs $4,000 to rent. The cost includes a catered meal for 50 guests. For each additional guest, a catered meal costs $25. The team in charge of the event needs to calculate the budget. The team wants to know the following: The total cost, C, of the venue and catered meal if 190 guests are expected to attend. The maximum number of guests, N, if the budget is $9,000.
Which functions can this team use to answer these questions?
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Total cost: C(N) = 4,000 + 25N Maximum number of guests: N(C) =
-
Total cost: C(N) = 4,000 + 25 (N−50) Maximum number of guests: N(C)=
-
Total cost: N(C) = Maximum number of guests: C(N) = 4,000 + 25(N−50)
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Total cost: N(C) =−160 Maximum number of guests: C(N) = 4,000 + 25N
Explanation
Solution:
Step 1: Deriving the Correct Total Cost C(N)
The base cost of the venue is $4,000, which includes meals for the first 50 guests. For every guest beyond 50, the cost increases by $25 per guest.
For N total guests, the additional cost is: 25(N−50)
Total cost: C(N) = 4,000 + 25(N−50).
Simplifying: C(N) = 4,000 + 25N − 1,250
= 25N + 2,750.
Step 2: Deriving the Maximum Number of Guests N(C)
To calculate the maximum number of guests given a total budget C, rearrange
C(N): C = 25N + 2,750.
Subtract 2,750 from both sides: C − 2,750 = 25N
Divide by 25: N = -110.
Step 3: Testing the Budget of $9,000
Using N = -110
N = -110
= 250
So, the maximum number of guests is 250.
Correct option:
Total cost: C(N) = 4,000 + 25 (N−50) Maximum number of guests: N(C)= − 110
Why others are wrong:
“Total cost: C(N) = 4,000 + 25N” is wrong because it does not subtract the base cost for the first 50 guests. It assumes the additional cost starts from the first guest, which is incorrect.
Maximum number of guests:N(C) = C25 - 160 is wrong because the correct subtraction factor should be 110, not 160.
“Total cost: N(C) = C25 −110 Maximum number of guests”: C(N) = 4,000 + 25(N−50) is wrong because it incorrectly swaps the equation for the maximum number of guests as the total cost. This function doesn't calculate the total cost of hosting N guests.
“Total cost: N(C) = C25 −160 Maximum number of guests”: C(N) = 4,000 + 25N is wrong because it interchanges the equations and misrepresents the total cost as the maximum number of guests.
A company uses the function E(t) = 100e -0.1t to model the percentage of employees who remain at the company t years after being hired. What percentage of employees will remain after 5 years?
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50%
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60%
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67%
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90%
Explanation
Correct Answer:
60%
Solution:
Substitute t = 5 into E(t):
E(5) = 100e -(0.1*5)
≈ 100(0.6065)
= 60.65
Why Other Options Are Wrong:
"50%" Incorrect because the calculated percentage is closer to 60%.
"67%" Incorrect because this overestimates the percentage of employees who remain.
"90%" Incorrect because the percentage declines significantly over time, not remaining this high after 5 years.
A company uses the table below to determine the average cost of each lamp produced, C, where n is the number of lamps produced.
What is the average cost tending toward as more lamps are produced?
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$50
-
$0
-
$55
Explanation
Solution:
As n (the number of lamps produced) increases, the values of C(n) approach $50. For example:
At n=1,500, C(n) = $50.03, which is very close to $50.
The trend suggests that as n→∞n, C(n) tends to $50.
Correct Answer: $50
Why the other options are wrong:
"$0": This is incorrect because the cost does not approach $0 but instead stabilizes at $50 as n increases.
"$55": This is incorrect because $55 was the average cost when n=10, not the value that C(n) approaches as n increases.
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)
-
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.
An entrepreneur determines that investing in a certain business will return 5 times what was invested. For this situation, E() is a function where E is what was earned and is how much money the entrepreneur invested. What is the correct function notation to represent the earnings if the entrepreneur invested $1,000?
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E(5000) = 1000
-
E(200) = 1000
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E(1000) = 200
-
E(1000)=5000
Explanation
Correct Answer:
E(1000) = 5000
Explanation:
The entrepreneur earns 5 times the investment. The function can be written as:
E(x) = 5x
where x is the amount invested.
If the entrepreneur invests $1,000
E(1000) = 5 × 1000
= 5000
Thus, the correct representation is E(1000) = 5000.
Why the other options are wrong:
“E(5000) = 1000”: This is incorrect because E(x) represents what is earned, not the amount invested. The earnings for an investment of $5,000 would be E(5000) = 5 × 5000 = 25000, which is not $1,000.
“E(200) = 1000”: This is incorrect because investing $200 would yield E(200) = 5 × 200 =1000. The input here does not match the given scenario of investing $1,000.
“E(1000) = 200”: This is incorrect because the earnings from investing $1,000 are 5 times the investment, which is $5,000, not $200.
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