An algorithm on a social media site is set up so that the more shares a video receives, the more individuals the video is presented to on their feed. The views of a particular video are increasing at an exponential rate of 100% for each share the video receives. If the video originated with 10 views, the situation can be modeled by the exponential equation v(s)=10(2)s

Question

where v is the number of views for every s shares. Which of the following statements is true about the equation?

(1 point)
Responses

The domain (s-values) is all real numbers.
The domain (s-values) is all real numbers. - no response given

The range (v-values) includes the values greater than (0<v≤10)
.
The range (v-values) includes the values greater than (0<v≤10)
. - incorrect

The domain (s-values) includes the values (s≥0)
.
The domain (s-values) includes the values open paren s is greater than or equal to 0 close paren. - no response given

The range (v-values) is all real numbers.

Answer 1

To analyze the equation \( v(s) = 10(2)^s \), we need to determine the valid domain and range of the function.

1. Domain:

   • \( s \) represents the number of shares. Since you cannot have a negative number of shares in this context, the domain is \( s \geq 0 \).
   • Therefore, the statement "The domain (s-values) includes the values \( s \geq 0 \)" is true.

2. Range:

   • When \( s = 0 \): \( v(0) = 10(2^0) = 10 \).
   • As \( s \) increases, since \( 2^s \) grows exponentially, \( v(s) \) will also increase without bound. Hence, the range of \( v(s) \) starts from 10 and goes to infinity.
   • Therefore, the range is \( v \geq 10 \) (all values greater than or equal to 10).
   • The statements about the range including values less than 10 or being all real numbers are incorrect.

Conclusively, the correct statement is: The domain (s-values) includes the values \( s \geq 0 \).

Answer 2

Marilee is creating a garden in her yard. She needs to put up a fence to keep the animals out of it. She has 40 yards of fencing she can erect. This situation is modeled using the equation A=−w^2+20w
, where A
is the area of the fence and w is the width of the fence. If you were to set up a table to determine the measurements of the width and its corresponding area, what would be a reasonable scale for the domain?(1 point)
Responses

5
5 - no response given

1
1 - no response given

2
2 - incorrect

10

Answer 3

To set up a table for the domain using the width \( w \) in the equation \( A = -w^2 + 20w \), we need to determine sensible values for \( w \) based on the fencing constraint and the nature of the quadratic function.

1. Understanding the equation:

   • The equation is a downward-facing parabola, which describes a maximum area that can be achieved.
   • Since the total perimeter is 40 yards of fencing, we have the relationship for a rectangular garden \( P = 2l + 2w = 40 \).

2. Finding the maximum width:

   • If we rearrange the perimeter equation, we can express the length in terms of the width: \[ l = 20 - w \]
   • Thus, the width \( w \) must be positive, and the maximum width can be found by setting up bounds.
   • Since the width cannot exceed half the total perimeter (20 yards), valid values for \( w \) must be between \( 0 \) and \( 20 \).

3. Choosing a reasonable scale for the domain:

   • Given that valid values for \( w \) range from \( 0 \) to \( 20 \), a scale of \( 1 \) or \( 5 \) would provide useful measurements for \( w \) to observe the function's properties effectively.
   • Using a scale of \( 2 \) would yield fewer data points, making it less practical for detailed analysis. A scale of \( 10 \) would provide even fewer points.

Given these considerations, either a scale of 1 or 5 would be reasonable for the domain. Since you are looking for which option stands out, a scale of 5 would be an effective way to present the width in wider intervals while still producing relevant area calculations.

Thus, if the intention is to pick one answer, 5 is a reasonable scale for the domain.