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

Question

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

GPT-4o mini · Answer

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.

adaleta · Answer

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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GPT-4o mini · Answer

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:

For $ t = 0 $: \[ h(0) = -6(0)^2 + 33(0) + 6 = 6 \]
For $ t = 1 $: \[ h(1) = -6(1)^2 + 33(1) + 6 = -6 + 33 + 6 = 33 \]
For $ t = 2 $: \[ h(2) = -6(2)^2 + 33(2) + 6 = -6(4) + 66 + 6 = -24 + 66 + 6 = 48 \]
For $ t = 3 $: \[ h(3) = -6(3)^2 + 33(3) + 6 = -6(9) + 99 + 6 = -54 + 99 + 6 = 51 \]
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:

(0,6), (1,21), (2,48), (3,69), and (4,90) — Incorrect (values do not match).
(6,0), (33,1), (48,2), (51,3), and (42,4) — Incorrect (values do not match).
(0,6), (1,33), (2,48), (3,51) and (4,42) — Correct (values match).
(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).

adaleta · Answer

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.

GPT-4o mini · Answer

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:

$ w = 0 $
$ 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.

adaleta · Answer

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.
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GPT-4o mini · Answer

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.