Applied Algebra - C957

The total profit, P( x), in thousands of dollars from the production of x thousands of laptop computers is given by the graph below. For this graph, x must be greater than or equal to 5. How many laptops must be produced to maximize the profit?

10,000

A company's total monthly cost is given by the linear function C( x) = 11,000 + 550 x , where C is the total monthly cost and x is the number of units produced each month. The graph of the function is given below: What does the y-intercept represent in this situation?

If 0 items are produced, the monthly cost is $11,000.00.

A company has the following information on last quarter's sales (month 7 refers to July): MonthSales (in thousands of dollars)730383289333 What was the average rate of change over the entire quarter (from July to September)?

Sales increased an average of $15,000 per month.

A cable company, a satellite dish company, and a streaming service all offer TV viewing plans based on the number of channels, c, on the plan. The price of each plan is given below: Cable: P( c) = 3 c + 60Satellite dish: P( c) = 4 c + 35Streaming service: P( c) = 5 c + 50 Which company's cost per additional channel is most expensive?

Streaming service

The profit function P( x) gives a company's total profit (in millions of U.S. dollars) for x products produced (in thousands). The graph of P( x) is below: At which number of products produced does the company stop being profitable?

About 4,500 items

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?

About 6 years after the department store opened, the store earned a maximum revenue of about $1,600,000.00.

A server at a restaurant has realized that tips, T, in dollars are related to the number of hours, h, that have been worked on any given day. Tips can be modeled by the function T( h) = X. How can this function be interpreted?

If the server works h hours, the tips will be $ X.

The following table compares the number of employees hired and the number of employees dismissed over an 11-year period for a certain business. Year# Hired# Dismissed20075010200820102009305201021172011603020127015201340202014155201520620161022017107 What can be concluded from this table?

More employees were hired than dismissed each year.

Each point on the following graph represents the revenue returned from a project with a specified marketing budget. The y-axis measures the revenue in thousands of dollars, and the x-axis represents the budget in thousands of dollars. Which budget should new projects have at this company to maximize revenue?

New projects should have a budget of $6,000 since this returned a revenue of $20,000.

Two medical operations are being compared to determine which is better for patients. Below is a graph of the chance of infection after operation (in percentage) for the first option (shown with a solid curve) and for the second option (shown with a dashed curve). Both curves are shown over time since 1950, in years. As time has gone on, which is the better operation?

Operation 1 is better because the chance of infection has dropped to a steady 10% since around the year 2000.

The population of the United States in millions of people can be estimated with the function P( t) = 282(1.01) t, where t = 0 represents the population, P, in the year 2000, and t is measured in years. What does P(-2) represent?

P(-2) represents the population in 1998, which was approximately 276 million people.

The following graph shows the price of the stock of a company x months after December 2014. A program is set to automatically buy this stock on the first day of the month, provided the price is below $20. What month would it have first bought this stock, and would the investment have increased in value the following month?

September 2015; yes, the stock would have increased in value the next month.

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?

The OBR forecasts the growth will remain above 1% between 2016 and 2021.

A bank uses the function A( t) = 1000(1.023) t to calculate an account balance, A, after t years. What is the average rate of change of the account balance from t = 2 to t = 12?

The account will increase by $26.72, on average, each year.

The following graph shows the number of page views for a company's website over time (in months). Estimate and interpret the average rate of change for the company's page views between March and August.

The average rate of change was about -240 page views per month between these times, which means that the company's page views went down 240 on average each month between these times.

The following graph shows the profits of a company for the last fiscal year. Approximately how much have the company's profits increased over the last year?

30%

A factory with an assembly line produces automobiles and is limited by the capacity of its assembly line. The model follows the function below, where A is the number of automobiles produced per week, and w is the number of workers at a given time. What happens as additional workers between 10 workers and 40 workers are added?

Adding additional workers increases the number of automobiles produced faster and faster.

The graph of the function C below represents the number of coal miners in the United States from 1985 to 2015.C( y) represents the number of coal miners in the United States in the year y. For example, C(1985) shows there were 170,000 coal miners in the United States in 1985. Which interpretation of the function C is accurate?

C(2000) represents 70,000 coal miners in 2000.

The profit for a manufacturer of a certain product is given by the function P( x) = -4 x 3 + 12 x 2 + 16 x, where P is the total profit (in thousands of dollars) for x items produced (in thousands). What is the average rate of change from x = 4 to x = 8?

Profit decreases by $288,000 per 1,000 items produced.

An airplane was cruising at an altitude of 39,000 feet and descended as shown in the table below. Descent of Airplane Altitude of Airplane (ft.)Time (min.)39,000033,0001027,0002021,0003015,000409,000503,00060 What can be concluded from this table?

The airplane had a descent rate of 600 ft./min.

A chain of stores started out as 100 stores the first year, then increased by 400 stores the following year, and then increased each year for the next six years. The function below tracks the growth of the chain during the first eight years. Which statement about the average rate of change is correct?

The average rate of change from years 1 to 4 is less than the average rate of change for years 4 to 7.

The function C( x), modeling cell phone price versus supply, is a decreasing logistic model (where C is the cost, and x is the number of cell phones available) that can be seen by the graph below. What is happening at x = 100?

The cost of the phone is decreasing faster and faster.

The unemployment rate in Poland for the time period from 1990 to 2012 is shown in the graph below using the function U( t) (measured as a percentage), where t is the year. For example, U(1990) is the unemployment rate for the year 1990. What can be concluded based on this graph?

The largest two-year increase in unemployment was from 1990 to 1992.

The volume of calls, V( h), at a particular customer service center can be written as a function of the number of hours after opening each day, h. What would the ordered pair (6,8) represent for this function?

There were 8 calls 6 hours after opening.

In the function C( x) = 21.25 + 0.25 x, C( x) represents the total cost, in dollars, for a passenger to ride a shuttle service for x miles. What is the total cost for a passenger to ride the shuttle for 50 miles?

$33.75

The following graph represents revenue from vending machines at a large business during a typical day. The y-axis represents revenue in dollars, and the x-axis represents time. What is the difference between the maximum revenue and the minimum revenue?

$85

A factory with an assembly line can produce more automobiles per week as more employees are working at a given time, limited by the capacity of the assembly line. The model follows the function below, where A is the number of automobiles per week, and w is the number of workers at a given time. How many workers are needed to produce 50 automobiles per week?

57 workers

A local coffee shop found that its number of customers, N, can be predicted by the time of day in hours, t. The opening time of 5:00 a.m. corresponds to t = 0. How should the graph be interpreted between t = 5.5 and t = 9?

Between 10:30 a.m. and 2:00 p.m., the number of customers is increasing faster and faster and then increasing slower and slower.

A company has developed two functions, L( t) and M( t), that determine the least amount and most amount, respectively, of revenue it will make in any given month in dollars (with t = 1 corresponding to January). What can the company reasonably conclude if L(2) = 3,000 and M(2) = 5,000?

In February, it will make between $3,000 and $5,000.

An appliance manufacturer has determined that the profit, P( x), that it makes in manufacturing and selling x dishwashers can be determined from the function in the graph below. How many dishwashers must the company manufacture and sell in order to make a profit of $80,000?

130

A company uses the function C = 15 e -n + 10, which is shown in the graph below, to model the average cost per toy produced, C, where n is the number of toys produced. How should this horizontal asymptote be interpreted?

As the number of toys produced increases, the average cost per toy tends toward $10.

The pricing models of four contractors are given below, where t is the number of hours worked and P is the total cost of the job. Contractor A: P( t) = 100 + 20 tContractor B: P( t) = 99 + 25 tContractor C: P( t) = 50 + 35 tContractor D: P( t) = 80 + 25 t Which contractor gives the cheapest hourly rate?

Contractor A

A bank begins a marketing campaign to increase its number of customers. Each month, all new customers are put into a drawing. One customer then receives a $50 credit in a checking account.The amount of money given away in the drawing, G, is a function of the number of months, M, the drawing has been going on for. What is the correct function notation to represent the amount of money given away after the drawing has been going on for 5 months?

G(5) = 250

A local business is looking to become a franchise. The company wants to predict the number of franchises, F, given a certain number of years, Y. This prediction is modeled by the function F( Y). What can be concluded by F(5) = 10?

In 5 years, there will be 10 open stores.

A scientist found a sample of carbon-14 and knows the graph below models the exponential decay of carbon-14, where t is time measured in years and y is the amount of carbon-14 left in the sample in grams. What would the average rate of change over the time interval of t = 100 to t = 700 represent, and would that rate be positive or negative?

It represents the annual change in the number of grams of carbon-14 between 100 and 700 years. The rate is negative.

The following graph shows the number of page views for a company's website over time (in months). During which month were there approximately 150 page views?

July

A rabbit population can be estimated with the function , where t = 0 represents the rabbit population, P, in the year 2010, and t is measured in years. What is the average rate of change of the rabbit population from 2020 to 2040?

The rabbit population is increasing by 0.002 rabbits per year during this time.

he original value of a company car bought in January of 2006 was $22,800. Below is a function table that tracks the value of the car V( y) as it depreciates. Depreciation of Company AutomobileYear (y)Value (V(y)) 2006$22,8002007$18,2402008$15,5042009$13,1782010$11,2012011$9,5212012$8,0932013$6,8792014$5,8472015$4,970 V(2007) = $18,240 is the value of the car from January to December of 2007. Which other conclusion can be reached from this table?

V( y) decreases the most from 2006 to 2007.

A company has tracked the relationship between the selling price of an item, x, and the revenue generated by the sales of that item, f. The company performs a polynomial regression on the data and models the data with the function f( x) = -24.34 x 2 + 336.21 x + 53.88 with r 2 = 0.93. The company uses this model to predict that if it charges $12 per item, it will earn a revenue of approximately $583.44. Is this conclusion valid and why?

Yes, it is a valid conclusion because f(12) = 583.44, r 2 is close to 1, and 12 is not far outside of the known data values.

A company examined cost data in an effort to model the relationship between number of items produced, x, and cost, C. The regression equation is C( x) = 0.52 x + 366 with a coefficient of determination of r 2 = 0.92. The company plans to use this function to determine the cost of producing 45 items. Is the company's plan appropriate?

Yes, it is appropriate because the coefficient of determination is close to 1, and this is an interpolation value.

A company has tracked the depreciating value of a company car over a 10-year period. The company performed a regression and modeled the value of the car, V, at year x with the function V( x) = 26,878.23 e -0.21x with r 2 = 0.98. Would it be appropriate to use this model to make a prediction for the value of the car when the car is 12 years old?

Yes, it would be appropriate because the data has a strong correlation, and 12 is within the range for accurate extrapolation.

A local university collects data related to its increasing enrollment for the past 19 years. The university sees enrollment was increasing quickly but is now leveling off. The university models this with the logistic function with r 2 = 0.96, where t is time in years, and P is enrollment in thousands at time t . The graph of the model is given below. Would it be appropriate for the university to use this model to predict enrollment five years from now (when t = 24)?

Yes, the data are strongly correlated, so this would be considered a somewhat safe extrapolation.

The following graph shows the sales of two popular items a company sells for a random sample of months. The company has chosen to fit the data with a linear model, and r 2 = 0.82. Is the linear function a good regression model for the data?

Yes, the data is strongly correlated, and it has a roughly constant rate of change.

A real estate agent wanted to develop a model for the number of days a rental property sits vacant based on how the rental price compares to the market value. The graph below shows the rental price compared to market value and the number of days the house sat vacant for 15 of the real estate agent's previous rental properties. The real estate agent fit the data with an exponential function, which has an r 2 value of 0.83. Is this exponential function a good regression model for the data?

Yes, there is a strong correlation, and the data points increase approximately by a constant ratio.

A technology firm in Erie, Pennsylvania, receives a contract to develop a computer program controlling the angle of a solar array on a daily basis. In order to develop the program, the firm first needs to model the hours of sunlight for every day of the year. The graphs below show the amount of hours of daylight for Erie, Pennsylvania, on December 21, March 21, June 21, and September 21. From December to March, the hours of daylight are increasing faster and faster, and from March to June, they are increasing slower and slower. Which graph would be an appropriate model for this firm to use?

answer in image

Black Monday occurred on October 19, 1987, and represents the single largest decline of the Dow Jones average in history.The function f represents a seven-month period during which Black Monday occurred. The function value f(1) represents the Dow Jones average in June of 1987. Which function value represents the Dow Jones average during the month when Black Monday occurred?

f(5)

The graph below represents the projected year-end assets, A( t), in the Social Security trust fund in trillions of dollars, where t represents the number of years since 2000. In the graph, t is defined between and including 0 to 50. At which value of t is the fund increasing the most rapidly?

About t = 0

A cat population in a small town can be modeled with a logistic function and estimated with the table below, where t = 0 represents the cat population, P, in the year 2000, and t is measured in years. tP(t) 124 25531034150 When does the cat population reach 110?

During 2003

A company pays its employees a base salary each month plus a commission, C, based on the number of sales, s, made by an employee during the month. An employee made 8 sales in the last month and received a commission of $1,600. How would this be represented in function notation?

C(8) = 1600

The number of gym memberships since January 1, 2015, for a particular gym is given by the function N( x) = 100 + 11 x, where N is the number of gym members x months after January 1, 2015. How can the slope be interpreted?

Each month, 11 members join the gym.

A high-interest loan company uses the function L( d) = 100 e 0.02 d to calculate late fees, L, for payments overdue by d days. What does L(67) represent?

L(67) is the amount due in late fees for a payment that is 67 days overdue, which would be $381.90.

A company has tracked the relationship between the number of units of an item that sold, x, (measured in thousands) and the total revenue earned from the item in dollars, A. The company decided to perform an exponential regression and found the function A( x) = 223.7 e 0.04x with r 2 = 0.76. The company uses this model to predict that when it sells 180,000 items, its revenue will be approximately $300,000.Is this a valid conclusion and why?

No, an accurate prediction cannot be made when x = 180 because it is too far from the known data values.

In last week's sales meeting, a supervisor at a company stressed that current sales are increasing, but the graph of the sales is concave down. She seemed to emphasize this repeatedly to encourage the sales team to change the curvature of the sales. What could happen to the company sales if the sales graph keeps the trend of being concave down?

Sales could continue increasing slower and slower, or sales could begin decreasing faster and faster.

The average cost in dollars, A( x), to produce x cell phone screen covers is given by the graph below. What happens to the average cost as more and more covers are produced?

The average cost reduces to about $2 per cover.

A company is trying to develop a computer algorithm to predict whether a customer will make a purchase of $100 or more in an online store based on the customer's browsing behavior and various characteristics about the customer. As the company collects more customer data, it updates its algorithm to make it more accurate. The graph below shows the algorithm's error rate for various amounts of customer data. Which conclusion can the company make about the error rate for this algorithm?

The error rate stabilizes and approaches an error rate of 2.5%.

The following graph represents profit on the y-axis (in thousands of dollars) based on production levels on the x-axis (in thousands of units). Each valley represents the decrease in profit after a major production expansion. After which production expansion do profits increase the fastest on average?

The expansion at 1,500 units

A factory with an assembly line can produce more automobiles per week as more employees are working at a given time, limited by the capacity of the assembly line. The model follows the function below, where A is the number of automobiles per week, and w is the number of workers at a given time. How should the upper horizontal asymptote be interpreted?

The factory can produce a maximum of approximately 95 automobiles per week.

The function table below shows a company's revenue function (R) and expense function (E) for each month. In the second row, R(1) is the revenue total for January, and E(1) is the expense total for January. Monthly Revenues and Expenses Month (m)Revenues R(m)Expenses E(m)1 (Jan.)$2,000$2,2002 (Feb.)$3,220$3,9573 (Mar.)$2,579$2,5044 (Apr.)$4,100$1,6755 (May)$5,328$2,1226 (June)$6,745$1,9907 (July)$5,995$2,1118 (Aug.)$10,200$3,1839 (Sept.)$9,455$4,12910 (Oct.)$12,337$92711 (Nov.)$10,234$3,78612 (Dec.)$7,195$2,199 Which conclusion can be made about this function table?

The highest profit occurred during October.

The U.S. housing price index from 1970 to 2010 is plotted in the graph below. Which conclusion about this graph is correct?

The lowest housing price index occurred in 1970, and the highest housing price index occurred in 2005.

A student created the scatterplot of data on the graph below. The x-axis represents the number of items, and the y-axis represents the total cost of the product. The student then drew in a dotted line of best fit. The student's calculator produced r 2 = 0.9946991797. How can the value of r 2 be interpreted?

The number of items has a strong correlation to the total cost.

The salvage or depreciated value, V, of a new truck purchased for $58,000 is given by the linear function V( t) = 58,000 - 4,000 t, where t is the time in years between 0 and 10. What is the correct interpretation of V(3)?

The truck's depreciated value after 3 years is $46,000.

A factory with an assembly line can produce more automobiles per week as more employees are working at a given time, limited by the capacity of the assembly line. The model follows the function below, where A is the number of automobiles produced per week, and w is the number of workers at a given time. The instantaneous rate of change at w = 50 is 1. How should this be interpreted?

When the factory has 50 workers, the factory will produce 1 more car for each worker added.

Below is a growth graph of a business. It compares gross growth and net growth. What can be said about the growth between the years of 2012 and 2014?

Both gross and net growth increased.

A company has tracked the depreciating value of a company car over an eight-year period. The company performed a linear regression and modeled the value of the car, V, at year x with the function V( x) = -2606.54 x + 24109.37 with r 2 = 0.96. Would it be appropriate to use this model to predict the value of the car when the car is 10 years old?

No, it would not be appropriate because past x = 9 the linear trend will not continue, and data cannot be extrapolated.

The rate that a rumor spreads through a middle school can be determined using the graph below, where R is the number of people who have heard the rumor after d days of it being introduced to the student body. When should approximately 25% of the maximum number of people hear the rumor?

On day 3

A software developer recently launched a new social media app and is concerned about user retention (i.e., users continuing to use the app after initial download). In order to study the retention of users, the company looked at users who shared three or more pictures on the app within the first five days of downloading the app (Group A, which is indicated by the solid line in the graph below) and compared the use of the app by these users in the days following the initial download with the use of the app in the days following the initial download by users who did not share three or more pictures on the app within the first five days after downloading the app (Group B, which is indicated by the dashed line in the graph below). The data in the graph shows the retention rates for these two groups, where t is the number of days since the initial download of the app, and y is the percentage of users who still use the app after t days. Which conclusion can be made from the data in this graph?

The retention rates of both Group A and Group B stabilize as time increases and appear to have different asymptotes.

An IT firm generates the following numbers to estimate its revenue in thousands of dollars based on the number of employees working on a project. EmployeesRevenue (in thousands)10$12.4020$21.9930$30.7440$36.0150$38.43 Which conclusion can be made by examining the average rate of change between consecutive rows?

The revenue is increasing at slower and slower rates in relation to the number of employees.

The following graph measures revenue from vending machines at a large business on a typical day. The y-axis shows revenue in dollars, and the x-axis shows time based on the 24-hour clock. Revenue was measured every 15 minutes from 7:00 a.m. (700) to 8:00 p.m. (2000). What indicates that a shift ends at 5:00 p.m. (1700)?

The vending machine revenues spike around 5:00 p.m. to about $55.

A big box store found its number of customers, C, in thousands on Saturday is predicted by the time of day in hours, t. In the graph below, t = 0 corresponds to midnight of the current day and t = 24 corresponds to midnight of the next day. How should the average rate of change from t = 8 to t = 10 be interpreted?

There are, on average, 1,000 more customers per hour.

A business wants to model the relationship between the number of items sold for two similar products. The monthly sales data for 20 months of sales of the two items are shown in the graph below. The company fit its data with a linear model, which has an R-squared value of r 2 = 0.18. What does r 2 imply for the fit of the model?

This model has a weak fit; therefore, predictions should not be made based on this function.