Question
Question 1
A)
Latesha’s mother puts $85 in Latesha’s lunch account at school. Each day, Latesha uses $3 from her account for lunch. The table below represents this situation. Fill in the amount remaining in Latesha's account each day after she purchases lunch. DO NOT type the $ symbol in the blank.
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
(5 points)
Day
Amount Left
in Account ($)
0 $85
1 $$$
2 $$$
3 $$$
4 $$$
5 $$$
Question 2
A)(4 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
The soccer team collected $800 at a car wash fundraiser. They charged $5 for small vehicles and $10 for larger vehicles. If x represents the number of small cars and y represents the number of large cars, the amount collected cal be modeled by the equation:
Response areax+
Response areay=
Response area.
If the soccer team washed 50 large vehicles, that means they had to wash Response area small vehicles to meet their $800 total.
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
Question 3
A)
Tyrone is comparing mobile phone plans. The plans are as follows:
Plan A: $95 for unlimited talk time and text messaging.
Plan B: $0.10 per minute for talk and $0.05 per text message.
Plan C: $20 monthly fee plus $0.05 per minute for talk and $0.05 per text message.
Plan D: $45 monthly fee $0.10 per minute for talk and free text messages.
If Tyrone talks for 800 minutes each month and uses 1000 text messages, put the phone plans in order from least expensive (top) to most expensive (bottom).
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
(4 points)
Arrange responses in the correct order to answer the question. Select a response, navigate to the desired position and insert response at that position. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can be moved using the up and down arrow keys or by dragging with a mouse.
Question 4
A)
The volume, V, of a rectangular prism is determined using the formula V=lwh
, where l is the length, w is the width, and h is the height of the prism.
Using the formula, what is the width of a rectangular prism that has a volume of V=138.24 in3
, a height of h=9.6 in
and a length of l=3.2 in
? Round your answer to the nearest tenth.
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
(1 point)
Responses
0.2 in
0.2 in
4.5 in
4.5 in
414.7 in
414.7 in
46.1 in
46.1 in
Question 5
A)
Brandon graphed one line in a system of equations. The system has only one solution of (0, 3)
. Which equation could be the other line in the system of equations? Select the three that apply.
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
(3 points)
Responses
x+y=−3
x plus y is equal to negative 3
2x−3y=−9
2 x minus 3 y is equal to negative 9
y=−13x+1
y is equal to negative 1 third x plus 1
y=3
y is equal to 3
y=12x+3
y is equal to 1 half x plus 3
Question 6
A)
The equation y=12x+4
is graphed below. Which equation would intersect this line at the point (4, 6)
?
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
(1 point)
Responses
y=4x
y is equal to 4 x
y=6
y is equal to 6
y=6x
y is equal to 6 x
y=4
y is equal to 4
Question 7
A)(2 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
The system of equations y=−15x−6
and y=−2x+3
is shown on the graph below.
According to the graph, the solution to the system of equations is (Response area, Response area).
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
Question 8
A)
The table below shows the conversion of liters to liquid pints.
Liters (x) Liquid Pints (y)
5 10.55
7 14.77
9 18.99
11 23.21
Which equation represents the relationship between the number of liters, x, and the number of liquid pints, y?
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
(1 point)
Responses
y=x−5.55
y is equal to x minus 5 point 5 5
y=x+5.55
y is equal to x plus 5 point 5 5
y=2.11x
y is equal to 2 point 1 1 x
y=x2.11
y is equal to x over 2 point 1 1
Question 9
A)
A student creates a table to evaluate the equation y=5x+8
. The student begins the table as shown. Fill in the missing values for the table.
(3 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
x Function y
1 5(1)+8
13
2 5(2)+
Response area Response area
3 Response area 23
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
Question 10
A)
The perimeter of a rectangle is 36 inches. The equation for the perimeter of a rectangle is P=2l+2w
, where l represents the length and w represents the width. Fill in the table for the possible values for length and width.
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
(3 points)
Length (in.) Width (in.)
$$ 12
7 $$
$$ 9
Question 11
A)
The formula to convert degrees Fahrenheit to degrees Celsius is C=59(F−32)
.
Convert 50oC
to oF
.
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
(1 point)
Responses
20oF
20 to the o th power cap f
132oF
132 to the o th power cap f
10oF
10 to the o th power cap f
122oF
122 to the o th power cap f
Question 12
A)
What is the solution to the system of equations graphed below?
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
(2 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
(Response area, Response area)
Question 13
A)
Which system of equations shows a solution of (−3.5, 4)
?
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
(1 point)
Responses
Question 14
A)
Identify the equation for the red line and the blue line. Then, identify the solution to the system of equations.
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
(3 points)
Drop target 1
Drop target 2
Drop target 3
A)
Latesha’s mother puts $85 in Latesha’s lunch account at school. Each day, Latesha uses $3 from her account for lunch. The table below represents this situation. Fill in the amount remaining in Latesha's account each day after she purchases lunch. DO NOT type the $ symbol in the blank.
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
(5 points)
Day
Amount Left
in Account ($)
0 $85
1 $$$
2 $$$
3 $$$
4 $$$
5 $$$
Question 2
A)(4 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
The soccer team collected $800 at a car wash fundraiser. They charged $5 for small vehicles and $10 for larger vehicles. If x represents the number of small cars and y represents the number of large cars, the amount collected cal be modeled by the equation:
Response areax+
Response areay=
Response area.
If the soccer team washed 50 large vehicles, that means they had to wash Response area small vehicles to meet their $800 total.
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
Question 3
A)
Tyrone is comparing mobile phone plans. The plans are as follows:
Plan A: $95 for unlimited talk time and text messaging.
Plan B: $0.10 per minute for talk and $0.05 per text message.
Plan C: $20 monthly fee plus $0.05 per minute for talk and $0.05 per text message.
Plan D: $45 monthly fee $0.10 per minute for talk and free text messages.
If Tyrone talks for 800 minutes each month and uses 1000 text messages, put the phone plans in order from least expensive (top) to most expensive (bottom).
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
(4 points)
Arrange responses in the correct order to answer the question. Select a response, navigate to the desired position and insert response at that position. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can be moved using the up and down arrow keys or by dragging with a mouse.
Question 4
A)
The volume, V, of a rectangular prism is determined using the formula V=lwh
, where l is the length, w is the width, and h is the height of the prism.
Using the formula, what is the width of a rectangular prism that has a volume of V=138.24 in3
, a height of h=9.6 in
and a length of l=3.2 in
? Round your answer to the nearest tenth.
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
(1 point)
Responses
0.2 in
0.2 in
4.5 in
4.5 in
414.7 in
414.7 in
46.1 in
46.1 in
Question 5
A)
Brandon graphed one line in a system of equations. The system has only one solution of (0, 3)
. Which equation could be the other line in the system of equations? Select the three that apply.
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
(3 points)
Responses
x+y=−3
x plus y is equal to negative 3
2x−3y=−9
2 x minus 3 y is equal to negative 9
y=−13x+1
y is equal to negative 1 third x plus 1
y=3
y is equal to 3
y=12x+3
y is equal to 1 half x plus 3
Question 6
A)
The equation y=12x+4
is graphed below. Which equation would intersect this line at the point (4, 6)
?
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
(1 point)
Responses
y=4x
y is equal to 4 x
y=6
y is equal to 6
y=6x
y is equal to 6 x
y=4
y is equal to 4
Question 7
A)(2 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
The system of equations y=−15x−6
and y=−2x+3
is shown on the graph below.
According to the graph, the solution to the system of equations is (Response area, Response area).
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
Question 8
A)
The table below shows the conversion of liters to liquid pints.
Liters (x) Liquid Pints (y)
5 10.55
7 14.77
9 18.99
11 23.21
Which equation represents the relationship between the number of liters, x, and the number of liquid pints, y?
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
(1 point)
Responses
y=x−5.55
y is equal to x minus 5 point 5 5
y=x+5.55
y is equal to x plus 5 point 5 5
y=2.11x
y is equal to 2 point 1 1 x
y=x2.11
y is equal to x over 2 point 1 1
Question 9
A)
A student creates a table to evaluate the equation y=5x+8
. The student begins the table as shown. Fill in the missing values for the table.
(3 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
x Function y
1 5(1)+8
13
2 5(2)+
Response area Response area
3 Response area 23
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
Question 10
A)
The perimeter of a rectangle is 36 inches. The equation for the perimeter of a rectangle is P=2l+2w
, where l represents the length and w represents the width. Fill in the table for the possible values for length and width.
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
(3 points)
Length (in.) Width (in.)
$$ 12
7 $$
$$ 9
Question 11
A)
The formula to convert degrees Fahrenheit to degrees Celsius is C=59(F−32)
.
Convert 50oC
to oF
.
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
(1 point)
Responses
20oF
20 to the o th power cap f
132oF
132 to the o th power cap f
10oF
10 to the o th power cap f
122oF
122 to the o th power cap f
Question 12
A)
What is the solution to the system of equations graphed below?
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
(2 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
(Response area, Response area)
Question 13
A)
Which system of equations shows a solution of (−3.5, 4)
?
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
(1 point)
Responses
Question 14
A)
Identify the equation for the red line and the blue line. Then, identify the solution to the system of equations.
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
(3 points)
Drop target 1
Drop target 2
Drop target 3
Answers
GPT-4o mini
Let's go through each question one by one.
### Question 1 A)
Latesha starts with $85 and spends $3 daily. The amount remaining each day can be calculated as follows:
- Day 0: $85
- Day 1: $85 - $3 = $82
- Day 2: $82 - $3 = $79
- Day 3: $79 - $3 = $76
- Day 4: $76 - $3 = $73
- Day 5: $73 - $3 = $70
So, the table filled out would look like this:
```
Day Amount Left in Account
0 85
1 82
2 79
3 76
4 73
5 70
```
### Question 2 A)
The equation to model the total amount collected by the soccer team can be structured as follows:
- For small vehicles (x): $5 each
- For large vehicles (y): $10 each
The equation would typically look like:
\[ 5x + 10y = 800 \]
If the team washed 50 large vehicles:
\[ 10(50) = 500 \]
Then, to meet $800, they need to collect:
\[ 800 - 500 = 300 \]
Thus:
\[ 5x = 300 \]
\[ x = 60 \]
So the required inputs are:
- Response area x: 5
- Response area y: 10
- Response area total: $800
- Response area small vehicles: 60
### Question 3 A)
Calculating the costs for Tyrone's usage:
- **Plan A**: $95
- **Plan B**: $0.10 * 800 + $0.05 * 1000 = $80 + $50 = $130
- **Plan C**: $20 + (0.05 * 800) + (0.05 * 1000) = $20 + $40 + $50 = $110
- **Plan D**: $45 + (0.10 * 800) = $45 + $80 = $125
Ordering from least expensive to most expensive:
1. Plan A: $95
2. Plan C: $110
3. Plan D: $125
4. Plan B: $130
### Question 4 A)
Volume formula: \( V = lwh \)
Given:
- V = 138.24 in³
- h = 9.6 in
- l = 3.2 in
We need to find \( w \):
\[ w = \frac{V}{lh} = \frac{138.24}{3.2 \times 9.6} \]
Calculating:
\[ 3.2 \times 9.6 = 30.72 \]
\[ w = \frac{138.24}{30.72} ≈ 4.5 \]
### Question 5 A)
To have a solution of (0, 3), we need lines that intersect at that point. An equation that passes through (0, 3) would have a y-intercept of 3.
The equations that apply:
1. y = 3
2. Any equation that can pass through that point.
Thus, you might check:
- **x + y = -3** (Doesn't work)
- **2x − 3y = −9** (Doesn't work)
- **y = −13x + 1** (Doesn't work)
- **y = 3** (Works)
- **y = 12x + 3** (Works, because it crosses (0, 3))
### Question 6 A)
If the equation is \( y = 12x + 4 \), to intersect at \( (4, 6) \):
\[ y = 12(4) + 4 = 48 + 4 = 52 \]
This does NOT match.
To find the correct one:
- We need an equation that results in \( y = 6 \).
**The equation that intersects is likely**:
1. **y = 6**
### Question 7 A)
To find the solution according to the graph:
- You'll need to look directly at the plotted lines and where they intersect.
Assuming it intersects at \( (x, y) \), if provided, you'll fill it in.
### Question 8 A)
Finding the relationship:
Observing the output/input pairs shows it seems to follow:
\[ y \approx 2.11x \]
### Question 9 A)
For the function \( y = 5x + 8 \):
- If x = 1:
- y = 5(1) + 8 = 13
- If x = 2:
- y = 5(2) + 8 = 18
- If x = 3:
- y = 5(3) + 8 = 23
### Question 10 A)
The perimeter is given as \( P = 36 \) inches:
\( 36 = 2l + 2w \)
You can find:
- If \( l = 12 \) then \( w = 6 \)
- If \( l = 7 \) then \( w = 11 \)
- If \( l = 9 \) then \( w = 9 \)
### Question 11 A)
For conversion from Celsius to Fahrenheit \( F = \frac{9}{5}C + 32 \)
For \( 50°C \):
\[ F = \frac{9}{5}(50) + 32 = 90 + 32 = 122°F \]
### Question 12 A)
Plotting will determine solutions, assuming (x, y) based on observed coordinates.
### Question 13 A)
Similarly, look for equations that yield (-3.5, 4).
### Question 14 A)
The two equations can be investigated and solutions adjusted accordingly based on a visual graph intersecting at identified points.
If specifics were provided for certain equations or plots, I could provide further exact values!
### Question 1 A)
Latesha starts with $85 and spends $3 daily. The amount remaining each day can be calculated as follows:
- Day 0: $85
- Day 1: $85 - $3 = $82
- Day 2: $82 - $3 = $79
- Day 3: $79 - $3 = $76
- Day 4: $76 - $3 = $73
- Day 5: $73 - $3 = $70
So, the table filled out would look like this:
```
Day Amount Left in Account
0 85
1 82
2 79
3 76
4 73
5 70
```
### Question 2 A)
The equation to model the total amount collected by the soccer team can be structured as follows:
- For small vehicles (x): $5 each
- For large vehicles (y): $10 each
The equation would typically look like:
\[ 5x + 10y = 800 \]
If the team washed 50 large vehicles:
\[ 10(50) = 500 \]
Then, to meet $800, they need to collect:
\[ 800 - 500 = 300 \]
Thus:
\[ 5x = 300 \]
\[ x = 60 \]
So the required inputs are:
- Response area x: 5
- Response area y: 10
- Response area total: $800
- Response area small vehicles: 60
### Question 3 A)
Calculating the costs for Tyrone's usage:
- **Plan A**: $95
- **Plan B**: $0.10 * 800 + $0.05 * 1000 = $80 + $50 = $130
- **Plan C**: $20 + (0.05 * 800) + (0.05 * 1000) = $20 + $40 + $50 = $110
- **Plan D**: $45 + (0.10 * 800) = $45 + $80 = $125
Ordering from least expensive to most expensive:
1. Plan A: $95
2. Plan C: $110
3. Plan D: $125
4. Plan B: $130
### Question 4 A)
Volume formula: \( V = lwh \)
Given:
- V = 138.24 in³
- h = 9.6 in
- l = 3.2 in
We need to find \( w \):
\[ w = \frac{V}{lh} = \frac{138.24}{3.2 \times 9.6} \]
Calculating:
\[ 3.2 \times 9.6 = 30.72 \]
\[ w = \frac{138.24}{30.72} ≈ 4.5 \]
### Question 5 A)
To have a solution of (0, 3), we need lines that intersect at that point. An equation that passes through (0, 3) would have a y-intercept of 3.
The equations that apply:
1. y = 3
2. Any equation that can pass through that point.
Thus, you might check:
- **x + y = -3** (Doesn't work)
- **2x − 3y = −9** (Doesn't work)
- **y = −13x + 1** (Doesn't work)
- **y = 3** (Works)
- **y = 12x + 3** (Works, because it crosses (0, 3))
### Question 6 A)
If the equation is \( y = 12x + 4 \), to intersect at \( (4, 6) \):
\[ y = 12(4) + 4 = 48 + 4 = 52 \]
This does NOT match.
To find the correct one:
- We need an equation that results in \( y = 6 \).
**The equation that intersects is likely**:
1. **y = 6**
### Question 7 A)
To find the solution according to the graph:
- You'll need to look directly at the plotted lines and where they intersect.
Assuming it intersects at \( (x, y) \), if provided, you'll fill it in.
### Question 8 A)
Finding the relationship:
Observing the output/input pairs shows it seems to follow:
\[ y \approx 2.11x \]
### Question 9 A)
For the function \( y = 5x + 8 \):
- If x = 1:
- y = 5(1) + 8 = 13
- If x = 2:
- y = 5(2) + 8 = 18
- If x = 3:
- y = 5(3) + 8 = 23
### Question 10 A)
The perimeter is given as \( P = 36 \) inches:
\( 36 = 2l + 2w \)
You can find:
- If \( l = 12 \) then \( w = 6 \)
- If \( l = 7 \) then \( w = 11 \)
- If \( l = 9 \) then \( w = 9 \)
### Question 11 A)
For conversion from Celsius to Fahrenheit \( F = \frac{9}{5}C + 32 \)
For \( 50°C \):
\[ F = \frac{9}{5}(50) + 32 = 90 + 32 = 122°F \]
### Question 12 A)
Plotting will determine solutions, assuming (x, y) based on observed coordinates.
### Question 13 A)
Similarly, look for equations that yield (-3.5, 4).
### Question 14 A)
The two equations can be investigated and solutions adjusted accordingly based on a visual graph intersecting at identified points.
If specifics were provided for certain equations or plots, I could provide further exact values!