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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=−w2+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

10
10 - no response given

1
1 - incorrect

2
2 - no response given
Incorrect

5 answers

To determine a reasonable scale for the domain in the context of this problem, we need to consider the width \( w \) of the fence and how it relates to the area \( A \).

Given the total amount of fencing is 40 yards, the perimeter of the fenced area can be represented as \( 2w + 2l = 40 \), where \( l \) is the length. This can be rearranged to determine the maximum width. To find the maximum possible width if we designate one variable, we can express length \( l \) in terms of width \( w \):

\[ l = 20 - w \]

From that, we see that \( w \) can range from 0 to 20 yards (because if \( w \) were greater than 20, \( l \) would become zero or negative).

Now, if we consider the width \( w \) incrementally, we would typically want to choose a scale that allows us to create a table with enough data points that illustrate the relationship effectively.

A scale of 5 or 10 would give us clear increments without crowding the table with data points that are too close together. In context, choosing increments of 5 would allow us to generate the points (0, 5, 10, 15, 20), which provides a good spread with which to understand the relationship between width \( w \) and area \( A \).

Therefore, the most suitable response from the options given for the reasonable scale for the domain would be 5.

A volleyball is served from a height of 6 feet with an initial velocity of 33 feet per second. The situation is modeled using the function h=−6t2+33t+6
, where h is the height in feet and t is time in seconds. Using the function, which ordered pair represents the vertex?(1 point)
Responses

(2.75,51.375)
left parenthesis 2.75 comma 51.375 right parenthesis

(0,6)
left parenthesis 0 comma 6 right parenthesis

(51.375,2.75)
left parenthesis 51.375 comma 2.75 right parenthesis

(5.676,0)

To find the vertex of the quadratic function given by the formula \( h = -6t^2 + 33t + 6 \), we can use the vertex formula for a quadratic function in the standard form \( h(t) = at^2 + bt + c \). The vertex \( t \)-coordinate can be found using the formula:

\[ t = -\frac{b}{2a} \]

In this case, the coefficients are:

  • \( a = -6 \)
  • \( b = 33 \)

Substituting these values into the vertex formula gives:

\[ t = -\frac{33}{2(-6)} = \frac{33}{12} = 2.75 \]

Next, we need to find the corresponding height \( h \) at \( t = 2.75 \). We can substitute \( t = 2.75 \) back into the height equation:

\[ h = -6(2.75)^2 + 33(2.75) + 6 \]

Calculating \( (2.75)^2 \):

\[ (2.75)^2 = 7.5625 \]

Now substituting this back into the equation:

\[ h = -6(7.5625) + 33(2.75) + 6 \] \[ h = -45.375 + 90.75 + 6 \] \[ h = -45.375 + 96.75 \] \[ h = 51.375 \]

Thus, we find the vertex to be at the point \( (2.75, 51.375) \).

The correct ordered pair representing the vertex is:

(2.75, 51.375).

Nora is creating a dog run in her backyard for her dog, Max. She has enough materials to create a run with a perimeter of 120 feet. This is represented by the function A=−w2+60w
where A
is the area of the run and w is the width. Which of the following identifies the x-intercept(s)?(1 point)
Responses

(0,0)
and (60,0)
left parenthesis 0 comma 0 right parenthesis and left parenthesis 60 comma 0 right parenthesis

(60,0)
left parenthesis 60 comma 0 right parenthesis

(30,900)
and (60,0)
left parenthesis 30 comma 900 right parenthesis and left parenthesis 60 comma 0 right parenthesis

(0,0)

To find the x-intercept(s) of the function \( A = -w^2 + 60w \), we need to set the area \( A \) equal to zero and solve for the width \( w \).

Setting \( A = 0 \):

\[ 0 = -w^2 + 60w \]

This can be factored as follows:

\[ 0 = w(-w + 60) \]

Setting each factor equal to zero gives:

  1. \( w = 0 \)
  2. \( -w + 60 = 0 \) → \( w = 60 \)

Thus, the x-intercepts are when \( w = 0 \) and \( w = 60 \). The corresponding points represented as ordered pairs would be:

  • \( (0, 0) \)
  • \( (60, 0) \)

Hence, the x-intercepts are:

(0,0) and (60,0).