1.14 Function model construction and application
A small company wants to analyze their monthly sales data over the past year to predict future sales trends. The company has recorded the total sales for each month and provided the following data:
| Month | Total Sales (thousands dollars) |
|---|---|
| \text{Jan} | 15 |
| \text{Feb} | 18 |
| \text{Apr} | 25 |
| \text{May} | 31 |
| \text{Jun} | 35 |
| \text{Jul} | 39 |
| \text{Aug} | 38 |
| \text{Sep} | 30 |
| \text{Oct} | 27 |
| \text{Nov} | 20 |
| \text{Dec} | 16 |
Explain how to identify the problem.
Explain how to create a model.
Explain how to apply and analyze the problem.
Explain how the results can be interpreted.
Explain how the model can be verified.
Explain how the findings can be reported.
For each of the following functions, describe the transformation of the parent function y=x^2.
y=(x-2)^2
y=-x^2+3
y=2(x+1)^2-4
y=-3(x-1)^2+5
Given the parent function x^3, perform the following transformations and write the resulting function:
Reflect the graph across the x-axis and shift down 2 units.
Compress the graph horizontally by a factor of 3 and shift left 1unit.
Stretch the graph vertically by a factor of 4 and shift up 5 units.
The population of a small town is modeled by the function P(t), where t is the number of years since 2000. The population in 2000 was 5000 people, and it is known that the population increases by 200 people per year.
Construct a mathematical model for the population of the town.
A company produces boxes and each box has a height h and a square base with side length s. The volume of the box must be 1000 cubic units, and the height must be twice the side length of the base. Construct a mathematical model for the dimensions of the box.
A car rental company charges a flat fee of \$20 per day for renting a car, plus an additional \$0.15 per mile driven. Let C(m) represent the cost of renting a car for a day and driving m miles. Construct a mathematical model for the cost of renting a car.
A cylindrical can is designed to hold 500 cubic centimeters of liquid. The height of the can is three times the diameter of the base. Construct a mathematical model for the dimensions of the can.
The graph of a function y=f(x) is shown. Determine the equation of the function, assuming it is a transformation of the parent function y=x^2.
A function y=g(x) is a transformation of the parent function y=x^2. The vertex of the parabola is at the point (3, -2), and the graph is reflected across the x-axisand stretched vertically by a factor of 4. Write the equation of the function y=g(x).
The graph of a function y=h(x) is shown below. Determine the equation of the function, assuming it is a transformation of the parent function y=x^3.
The table below shows the number of visitors to a local museum during the first 6 months of the year. Use regression techniques to construct a quadratic model for the given data set.
| Month | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Visitors | 1500 | 2300 | 3100 | 2900 | 2100 | 1800 |
A car rental company has recorded the number of cars rented per day for a week. The data is given in the table as shown. Use regression techniques to construct a cubic model for the given data set.
| Day | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Cars Rented | 12 | 15 | 25 | 32 | 29 | 18 | 10 |
A factory has a two-tiered production cost structure. The cost of producing up to 100 units is \$12 per unit, and the cost of producing more than 100 units is \$10 per unit. Construct a piecewise-defined function model to represent the total cost of producing x units.
A store offers a discount on bulk purchases of a product. The regular price is \$5 per item for purchases of less than 20 items, \$4 per item for purchases of 20 to 49 items, and \$3 per item for purchases of 50 or more items. Construct a piecewise-defined function model to represent the total cost of purchasing x items.
The force of gravity between two objects is inversely proportional to the square of the distance between them. If the force of gravity is 100 \text{ N} when the distance between the objects is 5 \text{ m}, determine the function that models the force of gravity between these objects as a function of the distance.
The intensity of light (\text{I}) from a light source is inversely proportional to the square of the distance (d) from the source. A light source has an intensity of 200 lux when the distance is 2 meters.
Determine the function that models the intensity of light as a function of the distance from the source.
Calculate the intensity of light when the distance from the source is 4 meters.
A population of rabbits is modeled by the function P\left(t\right) = 1000(1.12)^t, where P represents the rabbit population, and t represents the time in years.
Use the mathematical model to determine the rabbit population after 5 years.
The height of a bouncing ball in meters is modeled by the function H\left(t\right) = 10 - 5t^2, where H represents the height of the ball, and t represents the time in seconds.
Use the mathematical model to determine the height of the ball after 1 second and 2 seconds.
The temperature, T(t), in degrees Celsius, of a cup of coffee t minutes after being poured is given by the function T(t) = 20 + 70e^{-0.1t}.
Estimate the temperature of the coffee 5 minutes after being poured.
Calculate the average rate of change between 3 and 6 minutes and explain what it represents.
A car's fuel consumption (C) is inversely proportional to its speed (s). When the speed is 60 \text{ km/h}, the fuel consumption is 10 \text{ L/$100$ km}.
Write an equation relating the car's fuel consumption (C) in \text{L/$100$ km} to its speed (s) in \text{km/h}.
Calculate the fuel consumption when the car is traveling at a speed of 80 \text{ km/h}. Include the correct unit of measure in your answer.
At what speed should the car travel to achieve a fuel consumption of 8 \text{ L/$100$ km}? Include the correct unit of measure in your answer.
The time (t) it takes for a worker to complete a task is inversely proportional to the number of workers (w) assigned to the task. When 5 workers are assigned, the task is completed in 3 hours.
Write an equation relating the time (t) in hours to the number of workers (w).
How long will it take to complete the task if 10 workers are assigned? Include the correct unit of measure in your answer.
How many workers are required to complete the task in 1.5 hours? Include the correct unit of measure in your answer.
A forestry company uses the mathematical model y=5e^{0.2x} to predict the population of a certain tree species in a forest, where y represents the number of trees and x represents the number of years since the beginning of the observation. The company collected the following data after 5 years:
| \text{Years} | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| \text{Actual Population} | 5 | 6 | 7.3 | 8.9 | 10.9 | 13.5 |
| \text{After $5$ years} | 5 | 6.1 | 7.4 | 9.2 | 11.3 | 14.1 |
Evaluate the validity of the model by comparing its predictions to the actual data.
A scientist proposes a mathematical model y=3x^2-4x+2 to describe the height y (in meters) of a projectile above the ground at time x (in seconds). The following table shows the actual height data from an experiment:
| \text{Time ($s$)} | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| \text{Actual Height ($m$)} | 2 | 1 | 6 | 17 | 34 | 57 |
| \text{Predicted height} | 2 | 1.5 | 5.8 | 17.3 | 34.2 | 56.8 |
Evaluate the validity of the model by comparing its predictions to the actual data.
An economist develops a mathematical model y=1500(1.03)^x to predict the annual salary y (in dollars) of a worker x years after being hired. The following table shows the actual salary data for a worker:
| Years | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Actual Salary (dollars) | 1500 | 1545 | 1591 | 1638 | 1687 | 1738 |
| Predicted salary | 1500 | 1550 | 1595 | 1640 | 1690 | 1735 |
Evaluate the validity of the model by comparing its predictions to the actual data.