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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.

I require:
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

also, it would make me very happy to let you know htat i using a Desmos calculator, so if theres anything that might help me with this graph to make it what it needs to be, i would appreciate it.

9 answers

Let's address each of your questions step by step, including the requirements for graphs.

Question 2

Situation Choice: The number of insects in an area increases by 2% every week.

Identifying Quantities:

  • Independent Variable: Time (weeks)
  • Dependent Variable: Number of insects

Constant Percent Change: Yes, the dependent variable (number of insects) grows by a constant percent (2%) relative to the independent variable (time).

Modeling Equation: \[ N(t) = N_0 \cdot (1.02)^t \] Where:

  • \( N(t) \) = Number of insects at time \( t \)
  • \( N_0 \) = Initial number of insects
  • \( t \) = Number of weeks

Example: If you start with 100 insects:

  • \( N(t) = 100 \cdot (1.02)^t \)

Table of Values: | Weeks (t) | Number of Insects (N) | |-----------|------------------------| | 0 | 100 | | 1 | 102 | | 2 | 104.04 | | 3 | 106.1208 | | 4 | 108.243216 |

Graphing: Using Desmos, you can plot the function \( y = 100 \cdot (1.02)^x \) which represents the growth of insects over time.

Question 3

Concept of Decay: Decay by a constant percent rate per unit interval means that the quantity decreases by a fixed percentage in each interval of time, not a fixed amount.

Example Scenario: A radioactive substance decays at a rate of 5% per day.

Decay Equation: \[ A(t) = A_0 \cdot (0.95)^t \] Where:

  • \( A(t) \) = Amount of radioactive substance after \( t \) days
  • \( A_0 \) = Initial amount of the substance
  • \( t \) = Time in days

Table of Values (Assuming \( A_0 = 100 \)): | Days (t) | Remaining Amount (A) | |----------|-----------------------| | 0 | 100 | | 1 | 95 | | 2 | 90.25 | | 3 | 85.7375 | | 4 | 81.2740625 |

Graphing: Plot the function \( y = 100 \cdot (0.95)^x \) in Desmos.

Question 5

Situation Choice: The total cost of buying bottles of water at $2.50 per bottle and a water jug at $20.

Identifying Quantities:

  • Independent Variable: Number of bottles (x)
  • Dependent Variable: Total Cost (y)

Modeling Equation: \[ y = 20 + 2.5x \]

Slope and Y-Intercept:

  • Slope (2.5): This represents the cost increase per additional bottle of water.
  • Y-Intercept (20): This represents the fixed cost of the water jug when no bottles are purchased.

Table of Values: | Number of Bottles (x) | Total Cost (y) | |------------------------|--------------------| | 0 | 20 | | 1 | 22.50 | | 2 | 25 | | 3 | 27.50 | | 4 | 30 |

Graphing: Plot the linear equation \( y = 20 + 2.5x \) in Desmos.

Desmos Tips:

  • Start with the graphing function by entering the equation directly into the input bar.
  • For tables, you can create a table of values in Desmos by clicking on the "+" sign and selecting "Table."
  • Adjust the axes if needed to clearly visualize the growth/decay of the quantities.

This should cover your questions completely, including how to set up the graphs in Desmos. If you have any further queries or need clarification, feel free to ask!

um simplify the tables so i know what goes hwere.

Certainly! Here's a simplified version of the tables for each question.

Question 2: Number of Insects

Modeling Equation: \( N(t) = 100 \cdot (1.02)^t \)

| Weeks (t) | Number of Insects (N) | |-----------|------------------------| | 0 | 100 | | 1 | 102 | | 2 | 104 | | 3 | 106 | | 4 | 108 |

Question 3: Radioactive Substance Decay

Modeling Equation: \( A(t) = 100 \cdot (0.95)^t \)

| Days (t) | Remaining Amount (A) | |----------|-----------------------| | 0 | 100 | | 1 | 95 | | 2 | 90.25 | | 3 | 85.74 | | 4 | 81.27 |

Question 5: Total Cost of Water Bottles

Modeling Equation: \( y = 20 + 2.5x \)

| Number of Bottles (x) | Total Cost (y) | |------------------------|------------------| | 0 | 20 | | 1 | 22.50 | | 2 | 25 | | 3 | 27.50 | | 4 | 30 |

This format should make it clearer where to place each value. If you need any further assistance, just let me know!

separate it into two parts featuring the numbers that go into each section of the table.

Sure! Here are the tables separated into two parts featuring the numbers that go into each section.

Question 2: Number of Insects

Modeling Equation: \( N(t) = 100 \cdot (1.02)^t \)

Data for the Table

  • Independent Variable (Weeks, \( t \)): 0, 1, 2, 3, 4
  • Dependent Variable (Number of Insects, \( N \)): 100, 102, 104.04, 106.12, 108.24

Complete Table

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

Question 3: Radioactive Substance Decay

Modeling Equation: \( A(t) = 100 \cdot (0.95)^t \)

Data for the Table

  • Independent Variable (Days, \( t \)): 0, 1, 2, 3, 4
  • Dependent Variable (Remaining Amount, \( A \)): 100, 95, 90.25, 85.74, 81.27

Complete Table

| Days (t) | Remaining Amount (A) | |----------|-----------------------| | 0 | 100 | | 1 | 95 | | 2 | 90.25 | | 3 | 85.74 | | 4 | 81.27 |

Question 5: Total Cost of Water Bottles

Modeling Equation: \( y = 20 + 2.5x \)

Data for the Table

  • Independent Variable (Number of Bottles, \( x \)): 0, 1, 2, 3, 4
  • Dependent Variable (Total Cost, \( y \)): 20, 22.50, 25, 27.50, 30

Complete Table

| Number of Bottles (x) | Total Cost (y) | |------------------------|------------------| | 0 | 20 | | 1 | 22.50 | | 2 | 25 | | 3 | 27.50 | | 4 | 30 |

This format highlights the various values going into each section of the tables for clarity. If you need further modifications or additional information, let me know!

when i try to sue your equations with the table it claims the 'N', forexample, is a function .

It seems like the issue arises from how the variables are defined in each equation and table. When using the equations with software or graphing tools (like Desmos), ensure that the variables are correctly set up in accordance with the inputs you're working with.

Clarifying Equations with Variables

Let's clarify the equations associated with their respective tables for better understanding:

Question 2: Number of Insects

Modeling Equation: \[ N(t) = 100 \cdot (1.02)^t \]

  • Here, \( N \) is the Number of Insects as a function of \( t \), the weeks.

Table of Values for Question 2:

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

Question 3: Radioactive Substance Decay

Modeling Equation: \[ A(t) = 100 \cdot (0.95)^t \]

  • Here, \( A \) is the Remaining Amount as a function of \( t \), the days.

Table of Values for Question 3:

| \( t \) (Days) | \( A(t) \) (Remaining Amount) | |----------------|-------------------------------| | 0 | 100 | | 1 | 95 | | 2 | 90.25 | | 3 | 85.74 | | 4 | 81.27 |

Question 5: Total Cost of Water Bottles

Modeling Equation: \[ y = 20 + 2.5x \]

  • Here, \( y \) is the Total Cost as a function of \( x \), the Number of Bottles.

Table of Values for Question 5:

| \( x \) (Number of Bottles) | \( y \) (Total Cost) | |------------------------------|----------------------| | 0 | 20 | | 1 | 22.50 | | 2 | 25 | | 3 | 27.50 | | 4 | 30 |

Suggestions for Using in Desmos

  1. For Growth (Insects):
    Use \( N(t) \) as the output for input \( t \):

    • Enter \( N(t) = 100 \cdot (1.02)^t \) or plot \( y = 100 \cdot (1.02)^x \) (replacing \( t \) with \( x \) for Desmos).

  2. For Decay (Radioactive Substance):
    Use \( A(t) \) as the output for input \( t \):

    • Enter \( A(t) = 100 \cdot (0.95)^t \) or plot \( y = 100 \cdot (0.95)^x \).

  3. For Total Cost:
    Use \( y \) as the total cost defined by the number of bottles:

    • Enter \( y = 20 + 2.5x \).

By ensuring you use consistent variable names (e.g., \( x \) and \( y \) rather than \( t \) and \( N \) in software environments), everything should function correctly in Desmos. If you have any other issues or further clarifications, just let me know!

okay, so just answer all the questiosn and crap IDK anymore -.-

No problem! Let’s go through each question clearly and concisely, providing answers that summarize everything you've asked for.

Question 1

  1. Identify and explain quantities:

    • Bacteria in a culture: Doubling every hour is exponential growth (not constant rate).
    • City population: Growing at a rate of 2% each year is also exponential growth (not constant rate).
    • Distance traveled by a car (60 miles/hour): This is linear growth, changing at a constant rate.
    • Money earned at $15.75/hour: This is also linear growth, changing at a constant rate.

  2. Chosen scenario: Distance traveled by a car going 60 miles per hour and the amount of money earned at $15.75 per hour show two quantities changing at a constant rate relative to one another.

    Mathematical Equation: \[ d = r \cdot t \] Where:

    • \( d \) = distance traveled
    • \( r \) = rate (60 miles/hour)
    • \( t \) = time in hours

  3. Real-world problem: If you drive for 3 hours, how far do you travel? \[ d = 60 \cdot 3 = 180 \text{ miles} \]

  4. Making predictions: For any given time \( t \), you can use \( d = 60t \) to predict distance traveled. Limitations include not accounting for variable speeds or stops.

Question 2

  1. Situation: Number of insects increases by 2% every week.

    • Independent Variable: Time (weeks)
    • Dependent Variable: Number of insects

    Constant Percent Growth: Yes

  2. Modeling Equation: \[ N(t) = N_0 \cdot (1.02)^t \]

  3. Table of Values (Assuming \( N_0 = 100 \)): | Weeks (t) | Number of Insects (N) | |-----------|------------------------| | 0 | 100 | | 1 | 102 | | 2 | 104.04 | | 3 | 106.12 | | 4 | 108.24 |

  4. Graph: Plot \( y = 100 \cdot (1.02)^x \) in Desmos.

Question 3

  1. Decay Concept: Decay at a constant percent rate means the quantity decreases by a fixed percentage each time interval.

  2. Example Scenario: A radioactive substance decaying at 5% per day.

    Modeling Equation: \[ A(t) = A_0 \cdot (0.95)^t \]

  3. Table of Values (Assuming \( A_0 = 100 \)): | Days (t) | Remaining Amount (A) | |----------|-----------------------| | 0 | 100 | | 1 | 95 | | 2 | 90.25 | | 3 | 85.74 | | 4 | 81.27 |

  4. Graph: Plot \( y = 100 \cdot (0.95)^x \) in Desmos.

Question 4

  1. Graph Elements: Identify the y-intercept and slope based on the graph you have.

  2. Linear Equation: If your graph looks linear, you might write something like: \[ y = mx + b \] (where \( m \) is the slope and \( b \) is the y-intercept).

  3. Real-World Situation: Consider a situation like distance vs. time or cost vs. quantity.

  4. Prediction: Use the graph to help predict future values of either variable.

Question 5

  1. Linear Relationships:

    • Distance traveled by a car at 40 miles/hour: Linear relationship.
    • Total cost of water bottles: Linear relationship.
    • Area of a rectangle: Linear relationship.
    • Volume of a rectangular prism: Not linear, dependent on three dimensions.

  2. Chosen Situation: Total cost of buying water bottles.

    Modeling Equation: \[ y = 20 + 2.5x \] Where:

    • \( y \) = Total Cost
    • \( x \) = Number of Bottles

  3. Table of Values: | Number of Bottles (x) | Total Cost (y) | |------------------------|------------------| | 0 | 20 | | 1 | 22.50 | | 2 | 25 | | 3 | 27.50 | | 4 | 30 |

  4. Graph: Plot \( y = 20 + 2.5x \) in Desmos.

Desmos Tips

  • Input Equations: For exponential equations, use the form \( y = a \cdot (b)^x \).
  • Input Tables: You can create tables directly in Desmos by adding a table from the "+" menu.

Feel free to adjust or ask for more specific details on any of these parts, and I'm here to help!