A small nest of wasps has an exponential growth rate of 13% per month. If the nest currently has 5,000 wasps, the situation can be modeled by the equation w(t)=5,000(1.13)t

Question

, where "w" is the number of wasps after time "t" months. Which of the following statements is true about the equation? (1 point)
Responses

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

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

As "t" increases, "w" increases slowly at first and then quickly.
As "t" increases, "w" increases slowly at first and then quickly.

As "t" increases, "w" increases quickly at first and then slowly.

Answer 1

The equation you provided, \( w(t) = 5,000(1.13)^t \), shows exponential growth, where the number of wasps increases by 13% each month.

To analyze the growth behavior of this equation:

• Initially, when \( t \) is small, the multiplicative factor \( (1.13)^t \) is close to 1, meaning the increase in the number of wasps is relatively small.
• As \( t \) increases, the value of \( (1.13)^t \) grows larger, causing \( w(t) \) to increase more rapidly.

Based on this observation, the correct statement about the equation is:

As "t" increases, "w" increases slowly at first and then quickly.

Answer 2

The cheerleaders at a football game launch T-shirts into the crowd from the back of a golf cart that is 2 feet off the ground. The T-shirts have an upward velocity of 30 feet per second. Using the function y=−16t2+30t+2
, which of the following correctly identifies the y-intercept and best explains its meaning? (1 point)
Responses

The y-intercept is at (0, 2). The shirt will reach the ground after 2 seconds from being launched.
The y-intercept is at (0, 2). The shirt will reach the ground after 2 seconds from being launched.

The y-intercept is at (0, 2). The shirts are launched at an initial height of 2 feet.
The y-intercept is at (0, 2). The shirts are launched at an initial height of 2 feet.

The y-intercept is at (0, 0). The shirts are launched from the ground.
The y-intercept is at (0, 0). The shirts are launched from the ground.

The y-intercept is at (2, 0). The shirts are launched at an initial height of 2 feet.
The y-intercept is at (2, 0). The shirts are launched at an initial height of 2 feet.

Answer 3

To analyze the function \( y = -16t^2 + 30t + 2 \), we can identify its y-intercept by evaluating it at \( t = 0 \):

\[ y(0) = -16(0)^2 + 30(0) + 2 = 2 \]

This tells us that the y-intercept is at the point \( (0, 2) \).

The meaning of this y-intercept is that it represents the initial height of the T-shirts at the moment they are launched, which is 2 feet off the ground.

Given this information, the correct response is:

The y-intercept is at (0, 2). The shirts are launched at an initial height of 2 feet.

Answer 4

Given the function notation for the explicitly defined sequence f(n)=2n−1
, explain how you would find the fifth term in the sequence, then find/give the fifth term of the sequence.

Answer 5

To find the fifth term of the explicitly defined sequence given by the function \( f(n) = 2n - 1 \), you can follow these steps:

1. Identify \( n \): For the fifth term of the sequence, \( n \) will be 5.

2. Substitute \( n \): Substitute \( n = 5 \) into the function.

3. Calculate: Compute the result.

Let's do this step-by-step:

1. Let \( n = 5 \).

2. Substitute into the function: \[ f(5) = 2(5) - 1 \]

3. Calculate: \[ f(5) = 10 - 1 = 9 \]

Thus, the fifth term of the sequence is 9.

Answer 6

Use the explicit formula An=4.8+(n−1)2
to find the 11th term in the arithmetic sequence.(1 point)
A11=

Answer 7

To find the 11th term of the arithmetic sequence given by the formula \( A_n = 4.8 + (n - 1) \cdot 2 \), follow these steps:

1. Identify \( n \): Here, we want the 11th term, so \( n = 11 \).

2. Substitute \( n \): Plug \( n = 11 \) into the formula.

3. Calculate: Perform the arithmetic.

Let's go through the steps:

1. Let \( n = 11 \).

2. Substitute into the formula: \[ A_{11} = 4.8 + (11 - 1) \cdot 2 \]

3. Calculate: \[ A_{11} = 4.8 + (10) \cdot 2 \] \[ A_{11} = 4.8 + 20 \] \[ A_{11} = 24.8 \]

Thus, the 11th term \( A_{11} \) is 24.8.

Answer 8

Question
Manuel has saved $253.32 so far. He plans to put $50 into his savings each month for the next year. How much money will Manuel have after a year of saving (assuming no interest earned in the account)? Note: there are 12 months in a calendar year, but to get to the amount after a full calendar year, you would be looking at the January following the year Manuel is saving for.(1 point)
$

Answer 9

To calculate how much money Manuel will have after a year of saving, we need to add his initial savings to the total amount he plans to save over the year.

1. Initial savings: $253.32
2. Monthly savings: $50
3. Number of months: 12

First, calculate the total amount saved over the year:

\[ \text{Total savings over a year} = \text{Monthly savings} \times \text{Number of months} = 50 \times 12 = 600 \]

Now, add this amount to the initial savings:

\[ \text{Total amount after one year} = \text{Initial savings} + \text{Total savings over a year} = 253.32 + 600 = 853.32 \]

Therefore, after one year of saving, Manuel will have $853.32.

Answer 10

Use the table to answer the question.

The sequences for f(x)=2⋅2x
and g(x)=10x+2
are shown. Which of the following statements is true?

(1 point)
Responses

Statement #1: 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 eventually surpasses the growth rate of g(x).
Statement #1: 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 eventually surpasses the growth rate of g(x) .

Statement #3: 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 #3: 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 #2: The growth rate of g(x) is greater than the growth rate of f(x) between the second term and the fifth term.

Answer 11

To analyze the growth rates of the functions \( f(x) = 2 \cdot 2^x \) and \( g(x) = 10x + 2 \), it's helpful to recognize the nature of each function:

• Function \( f(x) = 2 \cdot 2^x \) is an exponential function. Its growth rate increases rapidly as \( x \) increases since exponential functions grow faster as their input increases.

• Function \( g(x) = 10x + 2 \) is a linear function. Its growth is constant; it increases by the same amount (10) for each increment of \( x \).

Assessing the Statements

• Statement #1 suggests that the growth rate of \( g(x) \) is initially greater than \( f(x) \) but that \( f(x) \) will eventually surpass \( g(x) \). This is misleading because exponential functions eventually outpace linear functions, but they actually start smaller at lower values of \( x\).

• Statement #3 suggests the opposite, that \( f(x) \) starts greater and that \( g(x) \) will eventually surpass \( f(x) \). This is incorrect since \( f(x) \) is always going to grow faster than \( g(x) \) for larger values of \( x\).

• Statement #2 states that the growth rate of \( g(x) \) is greater than that of \( f(x) \) between the second term and the fifth term. To evaluate this statement, we would need to compute specific values for \( f(x) \) and \( g(x) \) for \( x = 0, 1, 2, 3, 4, 5 \) and verify if, during that interval, \( g(x) \) indeed exceeds \( f(x) \).

Conclusion

Given that:

• For small values of \( x \), \( g(x) \) may initially be greater than \( f(x) \).
• However, as \( x \) increases, \( f(x) \) will surpass \( g(x) \) since \( f(x) \) grows exponentially.

Thus, Statement #2 is the most accurate because it could potentially reflect the behavior at specific values of \( x \) before the exponential growth of \( f(x) \) takes over.

So, the correct response would be: Statement #2: The growth rate of g(x) is greater than the growth rate of f(x) between the second term and the fifth term.