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:
- 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.
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.
- From \( (2, 320) \): \[ 320 = ab^2 \]
- 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.