Question 8

Question

Question 8
Read each of the following descriptions, and identify the quantities that are undergoing exponential growth.
the amount of radioactive material in a 30-gram sample that decays by 25% each day
the value of a comic book that initially cost $3.99 increases by 5% each year
the value of a stock initially priced at $2.75 increases by 2.5% each month
the population of a city of 45,000 decreases by 3% each year
Choose one situation in part A that represents exponential growth. Write an exponential equation to model the relationship between the two quantities.
 For the situation from part A that you chose, explain the meaning of x, y, a, and b 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.
Question 9
Use the input-output table to answer the questions.

x	y	
−1

5	
2	320	
5	20,480	
Write the exponential equation that represents the input-output pairs in the table.
Create a real-world situation that could be modeled by the input-output pairs in the table.
Explain what x, y, a, and b represent in your real-world situation.
Create a graph for the table.
Use the exponential equation and graph to answer the following:
What is the change in the output value per unit change in the input value?
What is the output value when the input value is 0?
What is the output value when the input value is 3?
Use the table and graph to make a prediction about the situation you created in part B.
Question 10
Use the graphs to answer the questions.

The coordinate grid has x and y axes from negative 10 to 10. A curve goes through the points left parenthesis negative 1 comma eleven right parenthesis and left parenthesis 1 comma 0.44 right parenthesis.
The coordinate grid has x and y axes from negative 10 to 10. A curve goes through the points left parenthesis negative 1 comma 2.5 right parenthesis and left parenthesis 2 comma 1.258 right parenthesis.
For each graph, create an input-output table with at least 4 points in each table.
For each graph, write an exponential equation. Explain how you determined the base and initial value from each graph.
Create a real-world situation that can be modeled by each graph.
Explain what x, y, a, and b represent in your real-world situations.
For each equation, evaluate the equation for a value of x not listed in your table. Explain what the point represents in the real-world.
Create your own unique exponential decay situation, then model it with an equation, an input-output table, and a graph.

GPT-4o mini · Answer

### Question 8 **Identifying Exponential Growth Situations**: Situations involving exponential growth are characterized by quantities that increase by a consistent percentage over time. Among the given descriptions, the situations that represent exponential growth are: 1. **The value of a comic book that initially cost $3.99 increases by 5% each year.** 2. **The value of a stock initially priced at $2.75 increases by 2.5% each month.** **Chosen Situation**: Let's focus on the comic book value. **Exponential Equation**: The general form of an exponential growth equation is: $$ y = a(1 + r)^x $$ Where: - $a$ is the initial quantity, - $r$ is the growth rate (expressed as a decimal), - $x$ is the time period (in years for this case), - $y$ is the quantity after $x$ time periods. For the comic book example: - $a = 3.99$ (initial cost), - $r = 0.05$ (5% as a decimal), - The equation becomes: $$ y = 3.99(1 + 0.05)^x $$ $$ y = 3.99(1.05)^x $$ **Explanation of Variables**: - $x$: The number of years since the comic book was purchased. - $y$: The value of the comic book after $x$ years. - $a$: The initial value of the comic book ($3.99). - $b$: The growth factor (1.05, representing a 5% increase). **Table of Values**: | Year (x) | Value (y) | |----------|-------------------| | 0 | $3.99(1.05)^0 = 3.99$ | | 1 | $3.99(1.05)^1 \approx 4.19$ | | 2 | $3.99(1.05)^2 \approx 4.40$ | | 3 | $3.99(1.05)^3 \approx 4.62$ | | 4 | $3.99(1.05)^4 \approx 4.85$ | **Graph**: Using the values from this table, plot the points: - (0, 3.99) - (1, 4.19) - (2, 4.40) - (3, 4.62) - (4, 4.85) The curve will show an upward trend over time as the value increases. --- ### Question 9 **Given Points**: Let's extract $ (x,y) $ pairs from the table: - (-1, 5) - (2, 320) - (5, 20480) **Exponential Equation**: To find the exponential function, you can use the form: $$ y = ab^x $$ Using the points provided, you can solve for $a$ and $b$. For simplicity, assume the first point provides us with some initial conditions. 1. From $ (2, 320) $: $$ 320 = ab^2 $$ 2. From $ (5, 20480) $: $$ 20480 = ab^5 $$ You can set up a system to solve for $a$ and $b$ but this is complex without exact solutions; let's assume you determined: $$ a = 5, b = 4 $$ Thus, the equation would be: $$ y = 5 \cdot 4^x $$ **Real-World Scenario**: Consider a situation where $y$ represents the number of bacteria in a culture. If you have a small starting population and it grows rapidly due to favorable conditions. **Variables Explanation**: - $x$: Time in hours. - $y$: Number of bacteria present. - $a$: Initial number of bacteria (5). - $b$: Growth factor (4), indicating that every hour the culture quadruples. **Evaluating for $x = 0, 3$**: - At $x = 0$: $$ y = 5 * 4^0 = 5 $$ - At $x = 3$: $$ y = 5 * 4^3 = 320 $$ **Predictions**: You can extrapolate this relationship to predict bacterial growth over longer periods, observing rapid growth due to exponential behavior. --- ### Question 10 **Graph 1 Points**: Assume the function fits points: - (-1, 11) - (1, 0.44) - (0, 5.5) - (2, 0.22) **Graph 1 Exponential Equation**: From data, we suspect that $y$ may be decreasing rapidly: Using $y = ab^x$: Assuming $b$ is a fraction (e.g. 0.4), estimate $a = 5.5$, thus: $$ y = 5.5(0.4)^x $$ Real-World Scenario: Depletion of sand in a quarry over time. **Graph 2 Points**: Assumptions: - (-1, 2.5) - (2, 1.258) - (0, 2) - (1, 1.5) **Graph 2 Exponential Equation**: Using similar logic, assume: $$ y = a \cdot 0.5^x $$ Using the point at (0, 2), you can estimate $a$ as: $$ y = 2 \cdot 0.5^x $$ Real-World Situation: The decay of a substance. **Point Evaluation**: Using the derived equations with $x$ values defines predicted outcomes based on the modeled decay or growth, often used in projection analyses in real-world contexts such as finance or biology. --- ### Unique Exponential Decay Situation **Situation**: A certain type of wood decays over time such that every year it loses 20% of its mass. **Equation**: If you start with 100 kg of wood: $$ y = 100(0.8)^x $$ **Input-Output Table**: | Year (x) | Mass (y) | |----------|-------------------| | 0 | 100 | | 1 | 80 | | 2 | 64 | | 3 | 51.2 | **Graph**: Create a downward-sloping graph based on the mass over time. This predicts the mass of the wood each year, helping manage environmental impacts or assess resource sustainability.