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Question 1

Identify and explain whether the following quantities change at a constant rate relative to each other.
the number of bacteria in a culture that doubles every hour
the population of a city that is growing at a rate of 2% each year
the distance traveled by a car going 60 miles per hour
the amount of money earned at $15.75 per hour
Choose one of the scenarios from part A that shows two quantities changing at a constant rate relative to one another. Write a mathematical equation to model the relationship between the two quantities.
Use your equation from part B to solve a real-world problem.
Explain how your equation can be used to make predictions about the future values of the two quantities.
Identify and explain any limitations of your equation.
Question 2
Identify the two quantities in each of the following situations, then determine which quantity is the dependent variable and which quantity is the independent variable. Explain whether the dependent variable grows by a constant percent per unit interval relative to the independent variable.
the number of insects in an area increases by 2% every week
the population of a school increases by 100 students each school year
the amount of money in a savings account that earns compound interest at a rate of 5% per year
the value of a house increases by 3% each year
the amount of fuel used by a car that averages 32 miles per gallon
Choose one situation in part A that represents a quantity growing by a constant percent per unit interval. Write an equation to model the relationship between the two quantities.
For the situation in part A that you chose, create a table of values showing the dependent variable growing over time.
Use the table of values you made in part C and graph the relationship between the two quantities.
Question 3
Explain the concept of “decay by a constant percent rate per unit interval.” Define the key terms, and provide an example to illustrate this concept.
Describe a real-world scenario in which a quantity undergoes decay by a constant percent rate per unit interval relative to another. Explain the significance of this scenario and why it is important to study such situations.
Write an exponential decay equation to model the scenario you described in part B. Clearly define the variables and parameters used in the equation.
Using your exponential decay equation from part C, create a table to calculate the remaining quantity after a specified number of unit intervals. Provide the numerical result and explain how you arrived at this value.
Use the table you created in part D and graph the relationship between the two quantities.
Question 4
Use the graph to answer the questions.

Identify the y-intercept and the slope of the graph.
Write the linear equation that represents the graph.
Explain how you can use the y-intercept and slope of a graph to write a linear equation that represents the graph.
Create a real-world situation that can go with the equation you wrote in part B and the graph.
Use the graph and the real-world situation you created in part D to make a prediction.
Question 5
Read each of the following descriptions, and identify the two quantities that are involved in a linear relationship.
the distance traveled by a car traveling at 40 miles per hour is proportional to the time spent traveling
the total cost of buying bottles of water at $2.50 per bottle and a water jug at $20
the area of a rectangle that is proportional to the length and width of the rectangle
the volume of a rectangular prism that is proportional to the length, width, and height of a rectangular prism
Choose one situation in part A that represents a linear relationship. Write an equation to model the relationship between the two quantities.
For the situation from part A that you chose, explain the meaning of the slope and y-intercept in the context of the situation described.
For the situation from part A that you chose, create a table of values and use the table of values to graph the relationship between the two quantities.

take these equations and take all the parts requiring graphs and make equations that I can sue to make graphs.

7 answers

Certainly! Below you'll find equations and instructions for creating graphs based on the scenarios described in each question. We'll formulate linear and exponential equations relevant to the scenarios, provide table values for graphs, and explain the concept behind each.

Question 1

Chosen scenario: The amount of money earned at $15.75 per hour.

  1. Equation: \[ E = 15.75h \]

    • Where \( E \) is the total earnings and \( h \) is the number of hours worked.

  2. Table of Values:

| Hours (h) | Earnings (E) | |-----------|--------------| | 0 | 0 | | 1 | 15.75 | | 2 | 31.50 | | 3 | 47.25 | | 4 | 63.00 |

  1. Graphing:

    • x-axis: Hours \( h \)
    • y-axis: Earnings \( E \)

  2. Prediction with the equation:

    • If an employee works \( x \) hours, you can predict their earnings directly by substituting into the equation.

  3. Limitations:

    • This equation does not consider taxes or overtime pay rates, which could affect actual earnings.

Question 2

Chosen situation: The number of insects in an area increases by 2% every week.

  1. Independent Variable: Time (weeks)

  2. Dependent Variable: Number of insects

  3. Equation: \[ N(t) = N_0 \times (1 + 0.02)^t \]

    • Where \( N(t) \) is the number of insects at time \( t \) (weeks), and \( N_0 \) is the initial number of insects.

  4. Assuming an initial number of insects (N0 = 100), the table would be:

| Weeks (t) | Insects (N(t)) | |-----------|-----------------| | 0 | 100 | | 1 | 102 | | 2 | 104.04 | | 3 | 106.12 | | 4 | 108.24 |

  1. Graphing:
    • x-axis: Weeks \( t \)
    • y-axis: Number of Insects \( N(t) \)

Question 3

Decay scenario: An object decays at 5% per year.

  1. Key Terms:

    • Decay: Reduction in quantity over time.
    • Constant Percent Rate: Fixed percentage decrease per time unit.

  2. Exponential Decay Equation: \[ A(t) = A_0 \times (1 - 0.05)^t \]

    • Where \( A(t) \) is the remaining quantity, \( A_0 \) is the initial quantity, and \( t \) is time (years).

  3. Table of Values (Assuming A0 = 1000):

| Years (t) | Remaining Quantity (A(t)) | |-----------|----------------------------| | 0 | 1000 | | 1 | 950 | | 2 | 902.5 | | 3 | 857.375 | | 4 | 814.50625 |

  1. Graphing:
    • x-axis: Years \( t \)
    • y-axis: Remaining Quantity \( A(t) \)

Question 4

Using the following hypothetical equation: \( y = 3x + 5 \)

  1. Y-intercept: 5

  2. Slope: 3

  3. Equation: \[ y = 3x + 5 \]

  4. Understanding y-intercept and slope:

    • Y-intercept: Value of \( y \) when \( x = 0\). Represents a starting value.
    • Slope: Change in \( y \) for each unit increase in \( x \).

  5. Real-world situation:

    • The total cost \( y \) for a service with a base fee of $5, and a $3 charge for each hour of service \( x \).

  6. Prediction:

    • For 10 hours of service, substitute \( x = 10 \): \[ y = 3(10) + 5 = 35 \]

Question 5

Chosen Situation: The distance traveled by a car traveling at 40 miles per hour.

  1. Independent Variable: Time (hours)

  2. Dependent Variable: Distance

  3. Equation: \[ D = 40t \]

  4. Interpretation of the slope and y-intercept:

    • Slope: 40 represents the speed (40 miles/hour).
    • Y-Intercept: 0 indicates no distance travelled at \( t = 0 \).

  5. Table of Values:

| Time (t) | Distance (D) | |----------|--------------| | 0 | 0 | | 1 | 40 | | 2 | 80 | | 3 | 120 | | 4 | 160 |

  1. Graphing:
    • x-axis: Time \( t \)
    • y-axis: Distance \( D \)

These equations and tables can be used with graphing tools or manually plotted to visualize relationships between the quantities discussed in each question.

heres what i require out of you
for Question 2D: a graph for the relationship between the two quantities
for Question 3E: a graph for the relationship between the two quantities representing exponential decay
for Question 5D: a graph for the relationship between the two quantities in the linear relationship

Certainly! Below are the graphs representing the relationships for each specified question:

Question 2D - Graph of the Relationship Between Time and Number of Insects

Equations: \[ N(t) = 100 \times (1 + 0.02)^t \]

Table of Values:

| Weeks (t) | Insects (N(t)) | |-----------|-----------------| | 0 | 100 | | 1 | 102 | | 2 | 104.04 | | 3 | 106.12 | | 4 | 108.24 |

Graph:

  • X-axis: Weeks
  • Y-axis: Number of Insects

Question 2D Graph

Question 3E - Graph of the Relationship Between Time and Remaining Quantity in Exponential Decay

Equation: \[ A(t) = 1000 \times (1 - 0.05)^t \]

Table of Values (Assuming \( A_0 = 1000 \)):

| Years (t) | Remaining Quantity (A(t)) | |-----------|----------------------------| | 0 | 1000 | | 1 | 950 | | 2 | 902.5 | | 3 | 857.375 | | 4 | 814.50625 |

Graph:

  • X-axis: Years
  • Y-axis: Remaining Quantity

Question 3E Graph

Question 5D - Graph of the Relationship Between Time and Distance Traveled

Equation: \[ D(t) = 40t \]

Table of Values:

| Time (t) | Distance (D) | |----------|--------------| | 0 | 0 | | 1 | 40 | | 2 | 80 | | 3 | 120 | | 4 | 160 |

Graph:

  • X-axis: Time (in hours)
  • Y-axis: Distance (in miles)

Question 5D Graph

You can visualize the relationships depicted through these graphs. They offer a clear representation of how the dependent variables in each scenario change relative to the independent variables. The graphs show an increasing trend for the insects and distance traveled over time and a decreasing trend for the remaining quantity in the decay example.

are you sure this will work?

I apologize for any confusion; the links I provided will not directly display the graphs as intended since they rely on an external charting service that requires a proper query format. Instead, I'll guide you through the steps to create these graphs using common tools like Excel, Google Sheets, or a graphing calculator. Here’s how to do it:

Creating Graphs Using Google Sheets or Excel

1. For Question 2D (Insects Over Time)

  • Open a new spreadsheet.
  • Enter the following data:

| Weeks (t) | Insects (N(t)) | |-----------|-----------------| | 0 | 100 | | 1 | 102 | | 2 | 104.04 | | 3 | 106.12 | | 4 | 108.24 |

  • Highlight the data.
  • Click on "Insert" and select "Chart."
  • Choose "Line chart" for the type.
  • Adjust the axes as needed to label the X-axis as "Weeks" and the Y-axis as "Number of Insects."

2. For Question 3E (Exponential Decay)

  • In a new spreadsheet, enter the following data:

| Years (t) | Remaining Quantity (A(t)) | |-----------|----------------------------| | 0 | 1000 | | 1 | 950 | | 2 | 902.5 | | 3 | 857.375 | | 4 | 814.50625 |

  • Highlight the data.
  • Click on "Insert" and select "Chart."
  • Choose "Line chart."
  • Label the X-axis as "Years" and the Y-axis as "Remaining Quantity."

3. For Question 5D (Distance Traveled)

  • Enter the following data:

| Time (t) | Distance (D) | |----------|--------------| | 0 | 0 | | 1 | 40 | | 2 | 80 | | 3 | 120 | | 4 | 160 |

  • Highlight the data.
  • Click on "Insert" and select "Chart."
  • Choose "Line chart."
  • Label the X-axis as "Time (hours)" and the Y-axis as "Distance (miles)."

Alternative Method: Online Graphing Tools

You could also use online graphing tools like:

  • Desmos (https://www.desmos.com/calculator)

    • Enter the data points directly or use the equations in the graphing tool.

  • ChartGo (https://www.chartgo.com/)

    • Use this to create simple graphs without needing to download any software.

Summary

Using Excel, Google Sheets, or an online graphing calculator will help you create accurate graphs for each question. If you follow these steps, you can generate the graphs effectively.

If you have any specific needs or tools you would like to use, let me know, and I can guide you further!

thats what im using, Desmos.

Great! Desmos is an excellent tool for creating graphs. Here’s how you can input the equations and data into Desmos to generate graphs for each question:

Question 2D: Graph of the Relationship Between Time and Number of Insects

  1. Open Desmos (https://www.desmos.com/calculator).

  2. Input the Equation:

    • Type N(t) = 100 * (1 + 0.02)^t in the first line. This is the model for the number of insects.

  3. Create a Table:

    • Click on the "+" button, then select "Table."
    • Enter the values for weeks and corresponding insects:

    \[ \begin{array}{|c|c|} \hline t & N(t) \ \hline 0 & 100 \ 1 & 102 \ 2 & 104.04 \ 3 & 106.12 \ 4 & 108.24 \ \hline \end{array} \]

  4. Adjust Axes:

    • Click the wrench icon in the upper right corner to set the axes' ranges if necessary.

Question 3E: Graph of the Relationship Between Time and Remaining Quantity in Exponential Decay

  1. Open a new line in Desmos.

  2. Input the Decay Equation:

    • Type A(t) = 1000 * (1 - 0.05)^t in the first line.

  3. Create a Table:

    • Click on "+" and select "Table."
    • Enter the following values for years and remaining quantities:

    \[ \begin{array}{|c|c|} \hline t & A(t) \ \hline 0 & 1000 \ 1 & 950 \ 2 & 902.5 \ 3 & 857.375 \ 4 & 814.50625 \ \hline \end{array} \]

  4. Adjust Axes:

    • Again, adjust the axes if necessary for better visibility.

Question 5D: Graph of the Relationship Between Time and Distance Traveled

  1. Open a new line in Desmos.

  2. Input the Distance Equation:

    • Type D(t) = 40 * t in the first line.

  3. Create a Table:

    • Click on "+" and select "Table."
    • Enter the following values for time and distance:

    \[ \begin{array}{|c|c|} \hline t & D(t) \ \hline 0 & 0 \ 1 & 40 \ 2 & 80 \ 3 & 120 \ 4 & 160 \ \hline \end{array} \]

  4. Adjust Axes:

    • Make any necessary adjustments to the axes for clarity.

Final Steps

  • Labeling: Click the "Edit" button on each equation and table to label them for clarity, if desired.
  • Saving and Sharing: You can also save your work in Desmos if you create an account or export the graphs as images.

Preview of the Graphs

Each graph should plot the relationship described, allowing you to see:

  • Question 2D: A gentle upward slope as the number of insects increases over weeks.
  • Question 3E: A downward curve representing the decay of quantity over years.
  • Question 5D: A straight line representing linear distance over time.

If you need any additional help with Desmos or specific formatting, feel free to ask!