. Identify the two quantities in each of the following situations, then determine

Question

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
B. For each situation in part A that represents a quantity growing by a constant
percent per unit interval, write an equation to model the relationships
between the two quantities.
C. For each situation in part A, create a table of values showing the dependent
variable growing over time.
D. Use the table of values you made in part C and graph the relationships
between the two quantities.

Answer 1

Let's analyze each situation in part A, identifying the dependent and independent variables, determining if they grow by a constant percent, and setting up equations where applicable. After this, we will create a table of values to illustrate the relationships and then discuss graphing them.

A. Identification of Variables

1. The number of insects in an area increases by 2% every week

   • Independent Variable: Time (weeks)
   • Dependent Variable: Number of insects
   • Growth: Yes, 2% growth per week (constant percent).

2. The population of a school increases by 100 students each school year

   • Independent Variable: Time (school years)
   • Dependent Variable: Population of the school
   • Growth: No, this is linear growth (constant amount, not percent).

3. The amount of money in a savings account that earns compound interest at a rate of 5% per year

   • Independent Variable: Time (years)
   • Dependent Variable: Amount of money in savings
   • Growth: Yes, 5% growth per year (constant percent).

4. The value of a house increases by 3% each year

   • Independent Variable: Time (years)
   • Dependent Variable: Value of the house
   • Growth: Yes, 3% growth per year (constant percent).

5. The amount of fuel used by a car that averages 32 miles per gallon

   • Independent Variable: Distance driven (miles)
   • Dependent Variable: Amount of fuel used (gallons)
   • Growth: No, this is linear growth (constant amount per distance).

B. Modeling Relationships with Equations

1. Insects: \[ N(t) = N_0(1 + 0.02)^t \] where \(N_0\) is the initial number of insects and \(t\) is the time in weeks.

2. Savings Account: \[ A(t) = P(1 + 0.05)^t \] where \(P\) is the principal amount and \(t\) is the time in years.

3. House Value: \[ V(t) = V_0(1 + 0.03)^t \] where \(V_0\) is the initial value of the house and \(t\) is the time in years.

C. Table of Values

Let's create a table of values for the quantities that grow by a constant percent.

1. Insects Example (Let \(N_0 = 1000\)) | Time (weeks) | Number of Insects | |--------------|--------------------| | 0 | 1000 | | 1 | 1020 | | 2 | 1040.4 | | 3 | 1061.208 | | 4 | 1082.43216 |

2. Savings Account Example (Let \(P = 1000\)) | Time (years) | Amount of Money | |---------------|-------------------| | 0 | 1000 | | 1 | 1050 | | 2 | 1102.5 | | 3 | 1157.625 | | 4 | 1215.50625 |

3. House Value Example (Let \(V_0 = 200000\)) | Time (years) | Value of House | |---------------|-------------------| | 0 | 200,000 | | 1 | 206,000 | | 2 | 212,180 | | 3 | 218,545.4 | | 4 | 225,101.122 |

D. Graphing Relationships

To graph these relationships:

1. For the Insects Growth:

   • Plot the points from the Insects table with time on the x-axis and number of insects on the y-axis.

2. For the Savings Account Growth:

   • Plot the points from the Savings Account table with time on the x-axis and amount of money on the y-axis.

3. For the House Value Growth:

   • Plot the points from the House Value table with time on the x-axis and house value on the y-axis.

When graphed, all three functions (for insects, savings, and house value) should show an exponential growth pattern, while the situation involving constant additions (like the school population) would show a straight line, indicating linear growth.

This is how you can analyze the relationships in each situation and understand how they grow over time.

Answer 2

can you make it more simple and like a kid wrote it

Answer 3

Sure! Let’s break it down into simpler language and a kid-friendly style.

A. What’s Happening?

1. Insects:

   • Independent Variable: Time (how many weeks pass)
   • Dependent Variable: Number of insects (how many insects there are)
   • Growth: Yes, the number grows by 2% every week (it gets a little bigger each week).

2. School Population:

   • Independent Variable: Time (how many school years go by)
   • Dependent Variable: Number of students (how many kids are in school)
   • Growth: No, it just adds 100 students every year (a straight line, not percent).

3. Savings Account:

   • Independent Variable: Time (how many years)
   • Dependent Variable: Amount of money (how much money you have)
   • Growth: Yes, it grows by 5% each year (a little more every year).

4. Value of a House:

   • Independent Variable: Time (how many years)
   • Dependent Variable: Value of the house (how much it's worth)
   • Growth: Yes, it grows by 3% every year (it gets a bit more expensive).

5. Fuel Used by a Car:

   • Independent Variable: Distance (how far you drive)
   • Dependent Variable: Amount of fuel (how much gas you use)
   • Growth: No, it’s a straight line (you use the same amount per mile).

B. Writing Simple Equations

1. Insects:

   • To show how many insects there are, we can say: \[ \text{Number of Insects} = \text{Starting Number} \times (1 + 0.02)^{\text{Weeks}} \]

2. Savings Account:

   • For the money, we can say: \[ \text{Money} = \text{Starting Money} \times (1 + 0.05)^{\text{Years}} \]

3. House Value:

   • For the house, we can say: \[ \text{Value} = \text{Starting Value} \times (1 + 0.03)^{\text{Years}} \]

C. Making a Table

Let’s create some simple tables for the insects, savings, and house value.

1. Insects (Let’s say we start with 1000 insects): | Weeks | Number of Insects | |-------|-------------------| | 0 | 1000 | | 1 | 1020 | | 2 | 1040 | | 3 | 1061 | | 4 | 1082 |

2. Savings Account (Starting with $1000): | Years | Amount of Money | |-------|-----------------| | 0 | $1000 | | 1 | $1050 | | 2 | $1102.50 | | 3 | $1157.63 | | 4 | $1215.51 |

3. House Value (Starting with $200,000): | Years | Value of House | |-------|-----------------| | 0 | $200,000 | | 1 | $206,000 | | 2 | $212,180 | | 3 | $218,545 | | 4 | $225,101 |

D. Graphing the Relationships

1. For Insects:

   • Draw a graph with weeks on the bottom (x-axis) and the number of insects on the side (y-axis). Plot the points from the insects table!

2. For Savings:

   • Draw a graph with years on the bottom (x-axis) and the amount of money on the side (y-axis). Plot the points from the savings table!

3. For House Value:

   • Draw a graph with years on the bottom (x-axis) and the value of the house on the side (y-axis). Plot the points from the house table!

When we graph these, the lines for insects, savings, and house value will all go up like a curve because they grow faster over time!

And that's it! You can see how these things grow over time and how they change!