Annabel wants to make banana bread for the bake sale. She went to the market to buy bananas, and she noticed fruit flies after she returned home with her produce. The number of fruit flies grows at an exponential rate modeled by the equation f(t)=8(2)t , where f is the number of fruit flies after t days. Define an appropriate domain for the problem if Annabel bakes the banana bread 5 days after returning from the market.(1 point)

Question

Responses

0≤f≤5
0 less than or equal to f less than or equal to 5

0≤t≤5
0 less than or equal to t less than or equal to 5

0≤f≤256
0 less than or equal to f less than or equal to 256

0≤t≤256
0 less than or equal to t less than or equal to 256
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Answer 1

To define an appropriate domain for the problem, we first analyze the function given:

The function for the number of fruit flies is \( f(t) = 8(2)^t \), where \( t \) is the number of days after Annabel returned from the market.

Since Annabel bakes the banana bread 5 days after returning from the market, we will consider the time interval from when she returns home (which we can consider as day 0) up to day 5. Thus, the value of \( t \) can vary from 0 to 5.

Therefore, the appropriate domain for \( t \) is: \[ 0 \leq t \leq 5 \]

Among the options provided, the correct response is: 0 ≤ t ≤ 5

Answer 2

Callie entered an art contest in second grade and won a $1,000 scholarship. The money was invested in an account paying a 9% interest rate compounded annually. The situation can be modeled by the equation a(t)=1,000(1.09)t , where a is the amount in the account after t years. If Callie uses the scholarship 10 years later, determine which graph accurately displays the situation.(1 point)

Answer 3

To determine which graph accurately displays the situation described, we need to analyze the given equation for compound interest:

\[ a(t) = 1,000(1.09)^t \]

Here, \( a(t) \) represents the amount in the account after \( t \) years, where the initial amount is $1,000 and the interest rate is 9%, compounded annually.

After 10 years, we can plug \( t = 10 \) into the equation to find out how much money Callie will have:

\[ a(10) = 1,000(1.09)^{10} \]

Calculating \( (1.09)^{10} \):

\[ (1.09)^{10} \approx 2.36736 \]

Now calculating \( a(10) \):

\[ a(10) \approx 1,000 \times 2.36736 \approx 2367.36 \]

After 10 years, the amount accumulated in the account will be approximately $2,367.36.

Characteristics of the Graph

1. Exponential Growth: The graph should show exponential growth, starting from $1,000 and increasing over time.
2. End Value: At \( t = 10 \), the graph should reach around $2,367.36.
3. Curve Shape: The curve should become steeper as \( t \) increases, indicating acceleration in growth due to compounding.

Choosing the Correct Graph

In an exam or a given set of graphs, you would look for a graph that reflects these characteristics—starting from $1,000, showing exponential growth, and crossing around $2,367.36 at the 10-year mark.

If you have specific graph options provided, you would select the one that meets these criteria. However, since the actual graphs are not displayed in this text, I can only guide you through the understanding of what to look for.

So, in summary, locate a graph showing exponential growth starting at $1,000 and reaching approximately $2,367.36 at \( t = 10 \).

Answer 4

A certain population of beetles is increasing at a rate of 22% per month. If there are currently 350 beetles, the equation f(x)=350(1.22)x can be used to model the situation. Finish the chart and plot the points to determine which graph correctly displays the situation.(1 point)

Answer 5

The equation you provided to model the population of beetles is:

\[ f(x) = 350(1.22)^x \]

Where \( f(x) \) is the number of beetles after \( x \) months. Here, \( 350 \) is the initial population, and \( 1.22 \) indicates a growth rate of 22% per month.

To fill in a chart and understand how the population changes over several months, we can calculate \( f(x) \) for different integer values of \( x \) (from 0 to, say, 5 or more months). Here are the calculations for the first few months:

1. At \( x = 0 \): \[ f(0) = 350(1.22)^0 = 350(1) = 350 \]

2. At \( x = 1 \): \[ f(1) = 350(1.22)^1 = 350(1.22) \approx 427 \]

3. At \( x = 2 \): \[ f(2) = 350(1.22)^2 = 350(1.4884) \approx 520 \]

4. At \( x = 3 \): \[ f(3) = 350(1.22)^3 = 350(1.8106) \approx 633 \]

5. At \( x = 4 \): \[ f(4) = 350(1.22)^4 = 350(2.2040) \approx 771 \]

6. At \( x = 5 \): \[ f(5) = 350(1.22)^5 = 350(2.6929) \approx 943 \]

Now, summarizing the values in a chart:

| \( x \) (Months) | \( f(x) \) (Beetles) | |-------------------|----------------------| | 0 | 350 | | 1 | 427 | | 2 | 520 | | 3 | 633 | | 4 | 771 | | 5 | 943 |

Plotting the Points

To plot the points:

• At \( (0, 350) \)
• At \( (1, 427) \)
• At \( (2, 520) \)
• At \( (3, 633) \)
• At \( (4, 771) \)
• At \( (5, 943) \)

Characteristics of the Graph

What to look for in the graph:

• The graph should start at \( (0, 350) \) and have values increasing rapidly to \( (5, 943) \).
• The shape of the curve will be upward and steepening, representing the exponential growth of the beetle population.

If you have specific graph options, choose the one that reflects these points and the described characteristics of rapid growth.

Answer 6

The value of a cell phone decreases exponentially at a rate of 24% each year after its release date. If the initial value of the phone is $700, the situation can be modeled by the equation v(t)=700(0.76)^t , where v is the value of the cell phone t years since its release date. Interpret the graph of the equation and identify the true statement.

(1 point)
Responses

As t increases, v decreases slowly at first and then quickly.
As t increases, v decreases slowly at first and then quickly.

As t increases, v increases quickly at first and then slowly.
As t increases, v increases quickly at first and then slowly.

As t increases, v decreases quickly at first and then slowly.
As t increases, v decreases quickly at first and then slowly.

As t increases, v increases slowly at first and then quickly.
As t increases, v increases slowly at first and then quickly.
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Answer 7

To interpret the graph of the equation

\[ v(t) = 700(0.76)^t \]

where \( v \) is the value of the cell phone after \( t \) years, we note the following key points:

1. Exponential Decay: The function \( v(t) \) represents exponential decay because the base \( 0.76 \) (which is less than 1) indicates that the value is decreasing over time.

2. Initial Value: At \( t = 0 \), the initial value of the cell phone is \( v(0) = 700(0.76)^0 = 700 \).

3. Behavior Over Time:

   • As \( t \) increases (that is, as time passes), the value of the cell phone decreases.
   • The rate at which it decreases is not constant: initially, the value drops more quickly, and as time goes on, the drops become smaller and smaller.

Graph Interpretation

• At first (in the early years), the phone loses a significant portion of its value due to the 24% annual depreciation.
• As time goes on, the remaining value is smaller, so the absolute amount lost each year becomes less, even though it is still a percentage of the remaining value.

True Statement

Based on the interpretation, the most accurate description of the behavior of the function is:

As t increases, v decreases quickly at first and then slowly.

This matches the characteristics of exponential decay, where the value decreases rapidly initially but the rate of decrease slows down as time progresses. Therefore, the correct response is:

As t increases, v decreases quickly at first and then slowly.

Answer 8

Mason is completing a study for his psychology course. For the study, he begins with $20 and asks individuals on the street if they want the money or if they would like him to double it and give it to the next person. He continues to do this until someone takes the money. This situation can be modeled by the exponential equation m(p)=20(2)p , where m is the money and p is the number of people who opt to pass it on to the next person. Which of the following statements is true about the equation?

(1 point)
Responses

As p increases, m decreases slowly at first and then quickly.
As p increases, m decreases slowly at first and then quickly.

As p increases, m increases quickly at first and then slowly.
As p increases, m increases quickly at first and then slowly.

The domain (p-values) include the values greater than or equal to 20 (p≥20).
The domain ( p -values) include the values greater than or equal to 20 left parenthesis p greater than or equal to 20 right parenthesis .

The range (m-values) includes the values (m≥20)

Answer 9

To analyze the given equation and determine which statement is true, we’ll look at the function:

\[ m(p) = 20(2)^p \]

Analyzing the Situation

• Initial Value: When \( p = 0 \) (when no one has passed it on), the money is \( m(0) = 20(2)^0 = 20 \).
• Doubling Effect: For each additional person who decides to pass on the money, the amount doubles.

Therefore, with each increase in \( p \):

• At \( p = 1 \), \( m(1) = 20(2)^1 = 40 \)
• At \( p = 2 \), \( m(2) = 20(2)^2 = 80 \)
• At \( p = 3 \), \( m(3) = 20(2)^3 = 160 \)

Behavior of the Graph

• As \( p \) increases, \( m \) increases exponentially. This means that even with a small increase in \( p \), the value of \( m \) increases quickly because of the exponentiation by \( 2 \).
• The pattern here is that as you keep passing the money to the next person, the total amount significantly increases, resulting in very rapid growth.

Evaluating the Statements

Now, let's evaluate the provided statements:

1. "As p increases, m decreases slowly at first and then quickly."

   • This statement is false. \( m \) does not decrease; it increases.

2. "As p increases, m increases quickly at first and then slowly."

   • This statement is false because \( m \) does not start increasing quickly and then slow down; it continues to increase rapidly due to the exponential nature of the function.

3. "The domain (p-values) include the values greater than or equal to 20 (p ≥ 20)."

   • This statement is false. The domain should include all non-negative integers, starting from \( 0 \) (e.g., p can be 0, 1, 2,...). Therefore, its domain is \( p \geq 0 \).

4. "The range (m-values) includes the values (m ≥ 20)."

   • This statement is true. Since \( m(p) = 20(2)^p \) will always yield values \( m \geq 20 \) for any non-negative integer \( p \).

Conclusion

The correct response is:

The range (m-values) includes the values (m ≥ 20).

Answer 10

check all your answers

Answer 11

Let's review each statement and my previous evaluations regarding the equation \( m(p) = 20(2)^p \):

1. Statement: "As \( p \) increases, \( m \) decreases slowly at first and then quickly."

   • Evaluation: This statement is false. As \( p \) increases, \( m \) does not decrease; it increases exponentially.

2. Statement: "As \( p \) increases, \( m \) increases quickly at first and then slowly."

   • Evaluation: This statement is false. As \( p \) increases, \( m \) increases rapidly and continues to increase quickly, not slowing down significantly, because it increases exponentially.

3. Statement: "The domain \((p\text{-values})\) include the values greater than or equal to 20 \((p ≥ 20)\)."

   • Evaluation: This statement is false. The domain actually includes all non-negative integers \( (p \geq 0) \) because passing the money can start from nobody having passed it on (i.e., \( p = 0 \)). Therefore, \( p \) starts from 0, not 20.

4. Statement: "The range \((m\text{-values})\) includes the values \((m ≥ 20)\)."

   • Evaluation: This statement is true. Since \( m(p) = 20(2)^p \) for any \( p \geq 0 \), \( m \) will always be 20 or greater. For \( p = 0 \), \( m = 20 \), and as \( p \) increases, \( m \) gets larger, thus \( m \) will always be \( \geq 20 \).

Summary of Correctness:

• False: "As \( p \) increases, \( m \) decreases slowly at first and then quickly."
• False: "As \( p \) increases, \( m \) increases quickly at first and then slowly."
• False: "The domain \((p\text{-values})\) include the values greater than or equal to 20 \((p ≥ 20)\)."
• True: "The range \((m\text{-values})\) includes the values \((m ≥ 20)\)."

Conclusion

The only correct statement is:

• "The range (m-values) includes the values (m ≥ 20)."

Thank you for prompting me to verify the analysis; it appears my original evaluation was correct regarding the true statement. Let me know if you have any further questions or if there's anything else you'd like to discuss!