Use the graph to answer the question.

Question

The graph of f(x)=100⋅1.5x
and g(x)=100x+100
is shown. Which of the following statements is true?

(1 point)
Responses

Statement A: The growth rate of g(x) is greater than the growth rate of f(x) between approximately x=0
and x=4
.
Statement A: The growth rate of g(x) is greater than the growth rate of f(x) between approximately x is equal to 0 and x is equal to 4.

Statement B: While the growth rate of g(x) is initially greater than the growth rate of f(x), the growth rate of f(x) keeps increasing and, by x=2.5
, surpasses the growth rate of g(x).
Statement B: While the growth rate of g(x) is initially greater than the growth rate of f(x) , the growth rate of f(x) keeps increasing and, by x is equal to 2 point 5, surpasses the growth rate of g(x) .

Statement C: While the growth rate of f(x) is initially greater than the growth rate of g(x), the growth rate of g(x) keeps increasing and eventually surpasses the growth rate of f(x).
Statement C: While the growth rate of f(x) is initially greater than the growth rate of g(x) , the growth rate of g(x) keeps increasing and eventually surpasses the growth rate of f(x) .

Statement D: All above statements are incorrect.

Answer 1

To determine which statement regarding the growth rates of the functions \( f(x) = 100 \cdot 1.5^x \) and \( g(x) = 100x + 100 \) is true, we need to analyze the growth behaviors of both functions.

1. Exponential Growth Function \( f(x) \): The function \( f(x) = 100 \cdot 1.5^x \) is an exponential function. Exponential functions grow rapidly as \( x \) increases. Its growth rate (the derivative) increases as \( x \) increases because the rate of change of an exponential function is proportional to its value.

2. Linear Growth Function \( g(x) \): The function \( g(x) = 100x + 100 \) is a linear function. Linear functions grow at a constant rate. The growth rate (the derivative) of \( g(x) \) is constant at 100, meaning it does not change as \( x \) increases.

3. Comparison of Growth Rates:

   • At \( x = 0 \):
     • \( f(0) = 100 \cdot 1.5^0 = 100 \)
     • \( g(0) = 100 \cdot 0 + 100 = 100 \)
   • At \( x = 1 \):
     • \( f(1) = 100 \cdot 1.5^1 = 150 \)
     • \( g(1) = 100 \cdot 1 + 100 = 200 \)
   • At \( x = 2 \):
     • \( f(2) = 100 \cdot 1.5^2 = 225 \)
     • \( g(2) = 100 \cdot 2 + 100 = 300 \)
   • At \( x = 3 \):
     • \( f(3) = 100 \cdot 1.5^3 = 337.5 \)
     • \( g(3) = 100 \cdot 3 + 100 = 400 \)
   • At \( x = 4 \):
     • \( f(4) = 100 \cdot 1.5^4 = 506.25 \)
     • \( g(4) = 100 \cdot 4 + 100 = 500 \)

From the observations, we see:

• Between \( x = 0 \) and \( x = 4 \), \( g(x) \) has a higher value than \( f(x) \) for \( x = 1, 2, 3 \), but at \( x = 4 \), \( f(4) \) surpasses \( g(4) \).
• However, the growth rate of \( g(x) \) remains constant (100), while the value of \( f(x) \) increases quite rapidly as \( x \) increases.

Evaluation of Statements:

• Statement A: True for \( x \) just after 0, but not completely valid, as \( f(x) surpasses ( g(x) \) at some point.
• Statement B: Correct that \( g(x) \) is initially greater, and \( f(x) \) eventually surpasses \( g(x) \) but not exactly at \( x = 2.5 \).
• Statement C: Incorrect because it states \( f(x) \) has a greater growth rate, which isn't true; \( g(x) \) has a constant linear growth rate which is lower than the exponential growth rate of \( f(x) \).
• Statement D: It cannot be an option since there are true aspects in the others.

The most accurate conclusion is Statement B: the growth rate of \( f(x) \) indeed increases and by some point, surpasses that of \( g(x) \). However, the exact crossing point is above \( x = 2.5 \), so it can be considered not precisely accurate. Thus the best insight is that there is growth behavior difference which need further evaluation.

Conclusively, Statement B holds truth regarding the characteristics described but is slightly approximate.

Answer 2

There is a protected species of bird living in the desert. There are only 1,234 birds left, and they are decreasing at a rate of 20 percent each year. Find the recursive and explicit formula to model the situation as well as the population after 3 years. Explain your process, and show all your work. (2 points)

Answer 3

To model the population of the protected species of bird using both a recursive and explicit formula, we can follow these steps:

1. Define the Initial Population

Let \( P_0 \) represent the initial population of the birds. In this case, we have: \[ P_0 = 1234 \]

2. Determine the Rate of Decrease

The population is decreasing at a rate of 20% per year. This can also be expressed as retaining 80% of the population each year. Therefore, the population for the next year can be expressed as: \[ P_n = P_{n-1} \times (1 - 0.20) = P_{n-1} \times 0.80 \]

3. Create the Recursive Formula

Using the information derived from the above steps, we can write the recursive formula as: \[ P_n = P_{n-1} \times 0.80 \quad \text{for } n \geq 1 \] with the initial condition: \[ P_0 = 1234 \]

4. Create the Explicit Formula

The explicit formula can be derived from the recursive formula. The population after \( n \) years can be expressed as: \[ P_n = P_0 \times (0.80)^n \] Substituting the initial population \( P_0 \): \[ P_n = 1234 \times (0.80)^n \]

5. Calculate the Population After 3 Years

Now, we can find the population after 3 years using both formulas.

Using the Explicit Formula: \[ P_3 = 1234 \times (0.80)^3 \]

Calculating \( (0.80)^3 \): \[ (0.80)^3 = 0.512 \]

So, \[ P_3 = 1234 \times 0.512 \] \[ P_3 \approx 631.168 \]

Rounding to the nearest whole number, we get: \[ P_3 \approx 631 \]

Summary of Results

• Recursive Formula: \( P_n = P_{n-1} \times 0.80 \) for \( n \geq 1 \), with initial condition \( P_0 = 1234 \)
• Explicit Formula: \( P_n = 1234 \times (0.80)^n \)
• Population after 3 Years: Approximately 631 birds.

Thus, after 3 years, we expect there to be about 631 birds remaining from the original population of 1,234.