A grocery store advertises the following cost for oranges.

Question

Number of oranges 1 3 5 9 12
Cost $0.89 $2.67 $4.45 $8.01 $10.68
Can the cost of oranges be modeled by a linear function?

(1 point)
Responses

Yes, because the rate of growth is a variable rate of change.
Yes, because the rate of growth is a variable rate of change.

No, because the rate of growth is a constant rate of change.
No, because the rate of growth is a constant rate of change.

No, because the rate of growth is a variable rate of change.
No, because the rate of growth is a variable rate of change.

Yes, because the rate of growth is a constant rate of change

Answer 1

To determine whether the cost of oranges can be modeled by a linear function, we need to check if the rate of change (also known as the slope) is constant between the different quantities of oranges.

Let's calculate the cost per additional orange:

1. From 1 to 3 oranges: \[ \text{Cost} = 2.67 - 0.89 = 1.78 \quad \text{(Cost for 2 additional oranges)} \] So, the average cost per orange = \( \frac{1.78}{2} = 0.89 \)

2. From 3 to 5 oranges: \[ \text{Cost} = 4.45 - 2.67 = 1.78 \quad \text{(Cost for 2 additional oranges)} \] Average cost per orange = \( \frac{1.78}{2} = 0.89 \)

3. From 5 to 9 oranges: \[ \text{Cost} = 8.01 - 4.45 = 3.56 \quad \text{(Cost for 4 additional oranges)} \] Average cost per orange = \( \frac{3.56}{4} = 0.89 \)

4. From 9 to 12 oranges: \[ \text{Cost} = 10.68 - 8.01 = 2.67 \quad \text{(Cost for 3 additional oranges)} \] Average cost per orange = \( \frac{2.67}{3} \approx 0.89 \)

The average cost per additional orange does not remain constant across all increments, as we see different amounts for different ranges (1.78, 3.56, and the final calculation). Therefore, the cost is not growing at a constant rate.

Based on this analysis, the correct response is:

No, because the rate of growth is a variable rate of change.

Answer 2

A research assistant is breeding mice to use in the lab. The following table lists the number of mice at the end of each week.

Week 1 2 5 9 11
Number of mice 24 48 120 216 264
Can the number of mice be modeled by a linear function?

(1 point)
Responses

Yes, because the rate of growth is a constant rate of change.
Yes, because the rate of growth is a constant rate of change.

Yes, because the rate of growth is a variable rate of change.
Yes, because the rate of growth is a variable rate of change.

No, because the rate of growth is a constant rate of change.
No, because the rate of growth is a constant rate of change.

No, because the rate of growth is a variable rate of change.

Answer 3

To determine if the number of mice can be modeled by a linear function, we need to check if the rate of change (slope) between the weeks is constant. We'll calculate the differences in the number of mice and see if they are consistent.

Let's calculate the changes in the number of mice for the given weeks:

1. From Week 1 to Week 2: \[ \text{Change} = 48 - 24 = 24 \]

2. From Week 2 to Week 5: \[ \text{Change} = 120 - 48 = 72 \]

3. From Week 5 to Week 9: \[ \text{Change} = 216 - 120 = 96 \]

4. From Week 9 to Week 11: \[ \text{Change} = 264 - 216 = 48 \]

Now we summarize the changes:

• Week 1 to Week 2: +24
• Week 2 to Week 5: +72
• Week 5 to Week 9: +96
• Week 9 to Week 11: +48

Since these changes (24, 72, 96, 48) are not constant, the growth is not linear.

Thus, the correct response is:

No, because the rate of growth is a variable rate of change.

Answer 4

A grocery store advertises the following cost for lemons.

Number of lemons 1 3 5 9 12
Cost $0.89 $2.67 $4.45 $8.01 $10.68
Determine whether a linear function or an exponential function would best model the increase in the cost of lemons, and why.(1 point)
Responses

a linear function, because the rate of increase is a constant percent rate of change
a linear function, because the rate of increase is a constant percent rate of change

an exponential function, because the rate of increase is a constant percent rate of change
an exponential function, because the rate of increase is a constant percent rate of change

a linear function, because the rate of increase is a constant rate of change
a linear function, because the rate of increase is a constant rate of change

an exponential function, because the rate of increase is a constant rate of change

Answer 5

To determine whether a linear function or an exponential function best models the increase in the cost of lemons, we need to evaluate the rate of change in the cost.

1. Calculate the increase in cost between different quantities of lemons:

   • From 1 to 3 lemons: \[ \text{Cost} = 2.67 - 0.89 = 1.78 \quad \text{(for 2 lemons)} \]
   • From 3 to 5 lemons: \[ \text{Cost} = 4.45 - 2.67 = 1.78 \quad \text{(for 2 lemons)} \]
   • From 5 to 9 lemons: \[ \text{Cost} = 8.01 - 4.45 = 3.56 \quad \text{(for 4 lemons)} \]
   • From 9 to 12 lemons: \[ \text{Cost} = 10.68 - 8.01 = 2.67 \quad \text{(for 3 lemons)} \]

2. Determine if the growth is constant or variable:

   The amount added (1.78, 1.78, 3.56, 2.67) shows that the increases are not consistent. This suggests that the growth is not linear (constant rate of change).

3. Check for constant percent rate of change:

   Next, let's analyze whether the increases suggest a constant percentage increase:

• From 1 to 3 lemons: \[ \text{Percent increase} = \frac{1.78}{0.89} \times 100 = 200% \]

• From 3 to 5 lemons: \[ \text{Percent increase} = \frac{1.78}{2.67} \times 100 \approx 66.54% \]

• From 5 to 9 lemons: \[ \text{Percent increase} = \frac{3.56}{4.45} \times 100 \approx 80% \]

• From 9 to 12 lemons: \[ \text{Percent increase} = \frac{2.67}{8.01} \times 100 \approx 33.33% \]

The percentage changes are not constant either, which further indicates that neither linear nor exponential functions perfectly capture the price changes.

However, since we are tasked with choosing a model based on the options provided and considering we see variable increases without a constant ratio, the best conclusion we can draw is:

An exponential function, because the rate of increase is a constant percent rate of change.

Although the rates of increase are not perfectly constant, it looks more like an exponential growth trend based on how the larger increments behave, typically associated with growth that accelerates. However, please note that based on the specific calculations, this is somewhat debatable as well.

Answer 6

Which of the following scenarios can be modeled by an exponential function?(1 point)
Responses

the amount of a radioactive isotope that decays by one-half every 7 years
the amount of a radioactive isotope that decays by one-half every 7 years - no response given

the amount of money in an account that has withdrawals of $35 each month
the amount of money in an account that has withdrawals of $35 each month - incorrect

the amount of a radioactive isotope that decays by 10 grams every hour
the amount of a radioactive isotope that decays by 10 grams every hour - no response given

the height of a ball thrown upwards

Answer 7

To determine which scenarios can be modeled by an exponential function, we need to identify situations where the change occurs at a constant relative rate (percentage) over time.

1. The amount of a radioactive isotope that decays by one-half every 7 years:
   This is an example of exponential decay. The amount of the isotope decreases by a constant percentage (50%) over a fixed period (7 years). This scenario can be modeled by an exponential function.

2. The amount of money in an account that has withdrawals of $35 each month:
   This scenario depicts a linear decrease in the account balance because a fixed amount is withdrawn each month. Therefore, it cannot be modeled by an exponential function.

3. The amount of a radioactive isotope that decays by 10 grams every hour:
   This is another linear scenario. The amount decreases by a fixed amount (10 grams) every hour, not a percentage. So this cannot be modeled by an exponential function.

4. The height of a ball thrown upwards:
   The height of a ball thrown upwards after it is released follows a parabolic trajectory rather than exponential growth or decay, so this is not modeled by an exponential function.

Given the above evaluations, the only scenario that can be modeled by an exponential function is:

The amount of a radioactive isotope that decays by one-half every 7 years.

Answer 8

Which one of the following scenarios can be modeled by an exponential function?(1 point)
Responses

the number of weeds in a garden that decreases by 20 per hour when the garden owner pulls the weeds
the number of weeds in a garden that decreases by 20 per hour when the garden owner pulls the weeds

the number of weeds in a garden that decreases by 25% each week when the gardener pulls the weeds
the number of weeds in a garden that decreases by 25% each week when the gardener pulls the weeds

the number of weeds in a garden that decreases by a different amount each week depending on who pulls the weeds
the number of weeds in a garden that decreases by a different amount each week depending on who pulls the weeds

the number of weeds in a garden that decreases by 200 each week when the landscaper pulls the weeds

Answer 9

To determine which scenario can be modeled by an exponential function, we need to identify whether the change occurs at a constant relative rate (percentage) over time.

1. The number of weeds in a garden that decreases by 20 per hour when the garden owner pulls the weeds:
   This scenario implies a fixed amount (20 weeds) being removed per hour. This is a linear decrease, not exponential.

2. The number of weeds in a garden that decreases by 25% each week when the gardener pulls the weeds:
   This scenario involves a constant percentage decrease (25%) of the total number of weeds each week. This is characteristic of exponential decay, where the amount decreases at a constant relative rate. Therefore, this scenario can be modeled by an exponential function.

3. The number of weeds in a garden that decreases by a different amount each week depending on who pulls the weeds:
   Since the decrease varies and is not a constant percentage, this scenario cannot be modeled by an exponential function.

4. The number of weeds in a garden that decreases by 200 each week when the landscaper pulls the weeds:
   Similar to the first scenario, this describes a fixed amount (200 weeds) being removed each week, which represents linear decay, not exponential decay.

Based on this analysis, the scenario that can be modeled by an exponential function is:

The number of weeds in a garden that decreases by 25% each week when the gardener pulls the weeds.