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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 fencin...Asked by adaleta
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. Using the equation, what is domain when the maximum range is achieved?(1 point)
Responses
10 yards
10 yards
5 yards
5 yards
10 square yards
10 square yards
100 yards
, where A
is the area of the fence and w is the width of the fence. Using the equation, what is domain when the maximum range is achieved?(1 point)
Responses
10 yards
10 yards
5 yards
5 yards
10 square yards
10 square yards
100 yards
Answers
Answered by
adaleta
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. Which of the following uses the function to correctly identify the corresponding y-values to the given x-values of 0, 1, 2, 3, and 4?(1 point)
Responses
(0,6)
, (1,21)
, (2,48)
, (3,69)
, and (4,90)
left parenthesis 0 comma 6 right parenthesis , left parenthesis 1 comma 21 right parenthesis , left parenthesis 2 comma 48 right parenthesis , left parenthesis 3 comma 69 right parenthesis , and left parenthesis 4 comma 90 right parenthesis
(6,0)
, (33,1)
, (48,2)
, (51,3)
, and (42,4)
left parenthesis 6 comma 0 right parenthesis , left parenthesis 33 comma 1 right parenthesis , left parenthesis 48 comma 2 right parenthesis , left parenthesis 51 comma 3 right parenthesis , and left parenthesis 42 comma 4 right parenthesis
(0,6)
, (1,33)
, (2,48)
, (3,51)
and (4,42)
left parenthesis 0 comma 6 right parenthesis , left parenthesis 1 comma 33 right parenthesis , left parenthesis 2 comma 48 right parenthesis , left parenthesis 3 comma 51 right parenthesis and left parenthesis 4 comma 42 right parenthesis
(0,0)
, (1,27)
, (2,42)
, (3,45)
, and (4,36)
left parenthesis 0 comma 0 right parenthesis , left parenthesis 1 comma 27 right parenthesis , left parenthesis 2 comma 42 right parenthesis , left parenthesis 3 comma 45 right parenthesis , and left parenthesis 4 comma 36 right parenthesis
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, where h is the height in feet and t is time in seconds. Which of the following uses the function to correctly identify the corresponding y-values to the given x-values of 0, 1, 2, 3, and 4?(1 point)
Responses
(0,6)
, (1,21)
, (2,48)
, (3,69)
, and (4,90)
left parenthesis 0 comma 6 right parenthesis , left parenthesis 1 comma 21 right parenthesis , left parenthesis 2 comma 48 right parenthesis , left parenthesis 3 comma 69 right parenthesis , and left parenthesis 4 comma 90 right parenthesis
(6,0)
, (33,1)
, (48,2)
, (51,3)
, and (42,4)
left parenthesis 6 comma 0 right parenthesis , left parenthesis 33 comma 1 right parenthesis , left parenthesis 48 comma 2 right parenthesis , left parenthesis 51 comma 3 right parenthesis , and left parenthesis 42 comma 4 right parenthesis
(0,6)
, (1,33)
, (2,48)
, (3,51)
and (4,42)
left parenthesis 0 comma 6 right parenthesis , left parenthesis 1 comma 33 right parenthesis , left parenthesis 2 comma 48 right parenthesis , left parenthesis 3 comma 51 right parenthesis and left parenthesis 4 comma 42 right parenthesis
(0,0)
, (1,27)
, (2,42)
, (3,45)
, and (4,36)
left parenthesis 0 comma 0 right parenthesis , left parenthesis 1 comma 27 right parenthesis , left parenthesis 2 comma 42 right parenthesis , left parenthesis 3 comma 45 right parenthesis , and left parenthesis 4 comma 36 right parenthesis
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Answered by
adaleta
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 best analyzes the meaning of the x-intercept?(1 point)
Responses
The dog run must have a width between 0 and 60 feet.
The dog run must have a width between 0 and 60 feet.
The dog run must have a width between 0 and 120 feet.
The dog run must have a width between 0 and 120 feet.
The dog run must have a width between 0 and 900 feet.
The dog run must have a width between 0 and 900 feet.
The dog run must have a width between 0 and 30 feet.
, where A
is the area of the run and w is the width. Which of the following best analyzes the meaning of the x-intercept?(1 point)
Responses
The dog run must have a width between 0 and 60 feet.
The dog run must have a width between 0 and 60 feet.
The dog run must have a width between 0 and 120 feet.
The dog run must have a width between 0 and 120 feet.
The dog run must have a width between 0 and 900 feet.
The dog run must have a width between 0 and 900 feet.
The dog run must have a width between 0 and 30 feet.
Answered by
adaleta
Use the image to answer the question.
A line with an arrow at the top is plotted in quadrant 1 of a coordinate plane. The x-axis is labeled Hours, and the y-axis is labeled Amount in Dollars.
Using this graph of a linear equation as an example, create a real-world situation that the graph could model. In 3–5 sentences, explain what the graph is modeling.
(2 points)
Short Answer Rubric (2 points)
Points Awarded Criteria
2
The answer is very clear.
The answer is fully supported by details.
There may be a few errors in grammar, usage, or mechanics. Errors do not interfere with meaning.
1
The answer is somewhat clear.
Details are weak or not related.
There are some errors in grammar, usage, and mechanics. Errors interfere with meaning.
0 The question is not answered.
Skip to navigation
A line with an arrow at the top is plotted in quadrant 1 of a coordinate plane. The x-axis is labeled Hours, and the y-axis is labeled Amount in Dollars.
Using this graph of a linear equation as an example, create a real-world situation that the graph could model. In 3–5 sentences, explain what the graph is modeling.
(2 points)
Short Answer Rubric (2 points)
Points Awarded Criteria
2
The answer is very clear.
The answer is fully supported by details.
There may be a few errors in grammar, usage, or mechanics. Errors do not interfere with meaning.
1
The answer is somewhat clear.
Details are weak or not related.
There are some errors in grammar, usage, and mechanics. Errors interfere with meaning.
0 The question is not answered.
Skip to navigation
Answered by
GPT-4o mini
To find the domain when the maximum range is achieved for the given equation, we first need to understand the context of the problem.
The equation provided is \( A = -w^2 + 20w \), where \( A \) is the area and \( w \) is the width.
To determine the dimensions of the fence, we need to consider that Marilee has 40 yards of fencing, which forms the perimeter of a rectangle. If we denote the length as \( l \) and the width as \( w \), the perimeter \( P \) can be expressed as \( P = 2l + 2w = 40 \). Solving for \( l \), we find:
\[
l = 20 - w
\]
The area \( A \) of the rectangle can then be expressed as:
\[
A = l \times w = w(20 - w) = 20w - w^2
\]
This confirms that the equation \( A = -w^2 + 20w \) is indeed the quadratic function representing the area.
To find the maximum area of the garden, we can find the vertex of the quadratic equation given by \( A = -w^2 + 20w \). The vertex \( w \) of a quadratic in standard form \( A = ax^2 + bx + c \) occurs at:
\[
w = -\frac{b}{2a}
\]
Here, \( a = -1 \) and \( b = 20 \):
\[
w = -\frac{20}{2(-1)} = \frac{20}{2} = 10
\]
Therefore, the maximum area occurs when the width \( w \) is 10 yards.
Now, checking the domain, since \( w \) represents the width and must be a positive quantity that satisfies the constraint of the perimeter (and must also result in a positive length), the practical range of \( w \) is between 0 yards and 20 yards (exclusive of the endpoints since we cannot have an area of zero). Thus, the domain for \( w \) is:
\[
0 < w < 20
\]
Since we are asked specifically about the maximum range achieved, when \( w = 10 \), it corresponds to an area calculation:
\[
A(10) = -(10^2) + 20(10) = -100 + 200 = 100 \quad \text{square yards}
\]
Now, relating your choices to this information:
- 10 yards (this refers to the width for max area)
- 5 yards (not relevant to max area)
- 10 square yards (wrong; this is not the area)
- 100 yards (this doesn’t match the units needed)
Based on the analysis, the choice that directly corresponds to achieving maximum area is **10 yards** (the width).
So, the answer is **10 yards** — this represents the width that maximizes the area.
The equation provided is \( A = -w^2 + 20w \), where \( A \) is the area and \( w \) is the width.
To determine the dimensions of the fence, we need to consider that Marilee has 40 yards of fencing, which forms the perimeter of a rectangle. If we denote the length as \( l \) and the width as \( w \), the perimeter \( P \) can be expressed as \( P = 2l + 2w = 40 \). Solving for \( l \), we find:
\[
l = 20 - w
\]
The area \( A \) of the rectangle can then be expressed as:
\[
A = l \times w = w(20 - w) = 20w - w^2
\]
This confirms that the equation \( A = -w^2 + 20w \) is indeed the quadratic function representing the area.
To find the maximum area of the garden, we can find the vertex of the quadratic equation given by \( A = -w^2 + 20w \). The vertex \( w \) of a quadratic in standard form \( A = ax^2 + bx + c \) occurs at:
\[
w = -\frac{b}{2a}
\]
Here, \( a = -1 \) and \( b = 20 \):
\[
w = -\frac{20}{2(-1)} = \frac{20}{2} = 10
\]
Therefore, the maximum area occurs when the width \( w \) is 10 yards.
Now, checking the domain, since \( w \) represents the width and must be a positive quantity that satisfies the constraint of the perimeter (and must also result in a positive length), the practical range of \( w \) is between 0 yards and 20 yards (exclusive of the endpoints since we cannot have an area of zero). Thus, the domain for \( w \) is:
\[
0 < w < 20
\]
Since we are asked specifically about the maximum range achieved, when \( w = 10 \), it corresponds to an area calculation:
\[
A(10) = -(10^2) + 20(10) = -100 + 200 = 100 \quad \text{square yards}
\]
Now, relating your choices to this information:
- 10 yards (this refers to the width for max area)
- 5 yards (not relevant to max area)
- 10 square yards (wrong; this is not the area)
- 100 yards (this doesn’t match the units needed)
Based on the analysis, the choice that directly corresponds to achieving maximum area is **10 yards** (the width).
So, the answer is **10 yards** — this represents the width that maximizes the area.
Answered by
GPT-4o mini
We start with the equation that models the height of the volleyball as a function of time:
\[
h = -6t^2 + 33t + 6
\]
We need to evaluate \( h(t) \) for the values of \( t = 0, 1, 2, 3, \) and \( 4 \). Let's calculate each one step by step:
1. **For \( t = 0 \)**:
\[
h(0) = -6(0)^2 + 33(0) + 6 = 6
\]
2. **For \( t = 1 \)**:
\[
h(1) = -6(1)^2 + 33(1) + 6 = -6 + 33 + 6 = 33
\]
3. **For \( t = 2 \)**:
\[
h(2) = -6(2)^2 + 33(2) + 6 = -6(4) + 66 + 6 = -24 + 66 + 6 = 48
\]
4. **For \( t = 3 \)**:
\[
h(3) = -6(3)^2 + 33(3) + 6 = -6(9) + 99 + 6 = -54 + 99 + 6 = 51
\]
5. **For \( t = 4 \)**:
\[
h(4) = -6(4)^2 + 33(4) + 6 = -6(16) + 132 + 6 = -96 + 132 + 6 = 42
\]
Now we summarize the results:
- \( (0, 6) \)
- \( (1, 33) \)
- \( (2, 48) \)
- \( (3, 51) \)
- \( (4, 42) \)
Now, we compare these results with the provided answer choices:
1. **(0,6), (1,21), (2,48), (3,69), and (4,90)** — Incorrect (values do not match).
2. **(6,0), (33,1), (48,2), (51,3), and (42,4)** — Incorrect (values do not match).
3. **(0,6), (1,33), (2,48), (3,51) and (4,42)** — Correct (values match).
4. **(0,0), (1,27), (2,42), (3,45), and (4,36)** — Incorrect (values do not match).
Thus, the correct response is:
**(0,6), (1,33), (2,48), (3,51), and (4,42)**.
\[
h = -6t^2 + 33t + 6
\]
We need to evaluate \( h(t) \) for the values of \( t = 0, 1, 2, 3, \) and \( 4 \). Let's calculate each one step by step:
1. **For \( t = 0 \)**:
\[
h(0) = -6(0)^2 + 33(0) + 6 = 6
\]
2. **For \( t = 1 \)**:
\[
h(1) = -6(1)^2 + 33(1) + 6 = -6 + 33 + 6 = 33
\]
3. **For \( t = 2 \)**:
\[
h(2) = -6(2)^2 + 33(2) + 6 = -6(4) + 66 + 6 = -24 + 66 + 6 = 48
\]
4. **For \( t = 3 \)**:
\[
h(3) = -6(3)^2 + 33(3) + 6 = -6(9) + 99 + 6 = -54 + 99 + 6 = 51
\]
5. **For \( t = 4 \)**:
\[
h(4) = -6(4)^2 + 33(4) + 6 = -6(16) + 132 + 6 = -96 + 132 + 6 = 42
\]
Now we summarize the results:
- \( (0, 6) \)
- \( (1, 33) \)
- \( (2, 48) \)
- \( (3, 51) \)
- \( (4, 42) \)
Now, we compare these results with the provided answer choices:
1. **(0,6), (1,21), (2,48), (3,69), and (4,90)** — Incorrect (values do not match).
2. **(6,0), (33,1), (48,2), (51,3), and (42,4)** — Incorrect (values do not match).
3. **(0,6), (1,33), (2,48), (3,51) and (4,42)** — Correct (values match).
4. **(0,0), (1,27), (2,42), (3,45), and (4,36)** — Incorrect (values do not match).
Thus, the correct response is:
**(0,6), (1,33), (2,48), (3,51), and (4,42)**.
Answered by
GPT-4o mini
To analyze the x-intercept of the function \( A = -w^2 + 60w \), we need to recognize that the x-intercept occurs when the area \( A \) is equal to zero. This means we can set the equation equal to zero and solve for \( w \):
\[
0 = -w^2 + 60w
\]
Rearranging gives us:
\[
w^2 - 60w = 0
\]
Factoring out \( w \):
\[
w(w - 60) = 0
\]
Setting each factor to zero gives us the following solutions:
1. \( w = 0 \)
2. \( w - 60 = 0 \) ⇒ \( w = 60 \)
The x-intercepts are therefore at \( w = 0 \) and \( w = 60 \).
Next, let's interpret what these values mean in the context of the dog run:
- When \( w = 0 \): This represents a width of zero, which means there is no dog run.
- When \( w = 60 \): This represents the maximum width such that the area still exists (above zero).
Since \( w \) represents the width, and we expect the width to be a positive quantity for the dog run, the feasible width for the dog run must lie between these two x-intercepts:
Thus, the viable width of the dog run must be **between 0 and 60 feet**.
Therefore, the correct interpretation of the x-intercept is:
**The dog run must have a width between 0 and 60 feet.**
\[
0 = -w^2 + 60w
\]
Rearranging gives us:
\[
w^2 - 60w = 0
\]
Factoring out \( w \):
\[
w(w - 60) = 0
\]
Setting each factor to zero gives us the following solutions:
1. \( w = 0 \)
2. \( w - 60 = 0 \) ⇒ \( w = 60 \)
The x-intercepts are therefore at \( w = 0 \) and \( w = 60 \).
Next, let's interpret what these values mean in the context of the dog run:
- When \( w = 0 \): This represents a width of zero, which means there is no dog run.
- When \( w = 60 \): This represents the maximum width such that the area still exists (above zero).
Since \( w \) represents the width, and we expect the width to be a positive quantity for the dog run, the feasible width for the dog run must lie between these two x-intercepts:
Thus, the viable width of the dog run must be **between 0 and 60 feet**.
Therefore, the correct interpretation of the x-intercept is:
**The dog run must have a width between 0 and 60 feet.**
Answered by
GPT-4o mini
The graph could represent the relationship between the number of hours worked and the total amount earned by a worker at a fixed hourly wage. For example, if an employee earns $15 per hour, the line would start at the origin (0 hours, $0) and rise consistently, reaching $15 after 1 hour, $30 after 2 hours, and so on. This model clearly shows that for every additional hour worked, the total earnings increase linearly by $15. The slope of the line reflects the hourly wage, while the x-axis indicates the hours worked and the y-axis represents the total earnings in dollars. This straightforward representation helps visualize how time directly correlates with income in this job scenario.