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Computer screens and television screens are usually measured by the diagonal of the screen. For example, a screen of size 60 in...Question
Computer screens and television screens are usually measured by the diagonal of
the screen. For example, a screen of size 60 inches would have a diagonal of 60
inches, a width of 52 inches, and a height of 30 inches.
Yoselin would like to install a new screen on a shelf that is 17 inches tall and 50
inches wide. She would like the screen to be the largest size that will fit on the
shelf. She knows the diagonal and width of the 4 screens listed below.
Screen Diagonal Width Height
A 15” 13”
B 22” 19”
C 32” 28”
D 40” 35”
the screen. For example, a screen of size 60 inches would have a diagonal of 60
inches, a width of 52 inches, and a height of 30 inches.
Yoselin would like to install a new screen on a shelf that is 17 inches tall and 50
inches wide. She would like the screen to be the largest size that will fit on the
shelf. She knows the diagonal and width of the 4 screens listed below.
Screen Diagonal Width Height
A 15” 13”
B 22” 19”
C 32” 28”
D 40” 35”
Answers
Answer
1. Use the Pythagorean Theorem to calculate the height of each screen, and
complete the table.
complete the table.
Answer
what are the directions in order to find these heights
Answer
Start with the Pythagorean Theorem. Plug in the sides that you know. Square 5 and 13.Subtract 25 from both sides. Take the square root of both sides of the equation
are these more understandable
are these more understandable
Answer
2. Which screen is the largest that will fit on Yoselin’s shelf?
Answered by
GPT-4o mini
To determine which screen will fit on Yoselin's shelf, we need to check both the height and the width of each screen against the dimensions of the shelf, which is 17 inches tall and 50 inches wide.
For screens, the relationship between the width (W), height (H), and diagonal (D) can be given by the Pythagorean theorem:
\[
D^2 = W^2 + H^2
\]
From this, we can rearrange it to find the height:
\[
H = \sqrt{D^2 - W^2}
\]
Let's calculate the height for each screen:
### Screen A
- **Diagonal (D)**: 15 inches
- **Width (W)**: 13 inches
\[
H = \sqrt{15^2 - 13^2} = \sqrt{225 - 169} = \sqrt{56} \approx 7.48 \text{ inches}
\]
### Screen B
- **Diagonal (D)**: 22 inches
- **Width (W)**: 19 inches
\[
H = \sqrt{22^2 - 19^2} = \sqrt{484 - 361} = \sqrt{123} \approx 11.09 \text{ inches}
\]
### Screen C
- **Diagonal (D)**: 32 inches
- **Width (W)**: 28 inches
\[
H = \sqrt{32^2 - 28^2} = \sqrt{1024 - 784} = \sqrt{240} \approx 15.49 \text{ inches}
\]
### Screen D
- **Diagonal (D)**: 40 inches
- **Width (W)**: 35 inches
\[
H = \sqrt{40^2 - 35^2} = \sqrt{1600 - 1225} = \sqrt{375} \approx 19.36 \text{ inches}
\]
### Comparing to the Shelf Dimensions
Now we will compare the height and width of each screen with the shelf dimensions (17 inches tall and 50 inches wide):
- **Screen A**:
- Height: 7.48 inches < 17 inches ✅
- Width: 13 inches < 50 inches ✅
- **Screen B**:
- Height: 11.09 inches < 17 inches ✅
- Width: 19 inches < 50 inches ✅
- **Screen C**:
- Height: 15.49 inches < 17 inches ✅
- Width: 28 inches < 50 inches ✅
- **Screen D**:
- Height: 19.36 inches > 17 inches ❌ (does not fit)
- Width: 35 inches < 50 inches ✅
### Conclusion
Screens A, B, and C fit the shelf, but screen D does not. Among the screens that fit, screen C has the largest diagonal of 32 inches. Therefore, the largest screen Yoselin can install on her shelf is:
**Screen C (32 inches diagonal, 28 inches wide, approximately 15.49 inches tall).**
For screens, the relationship between the width (W), height (H), and diagonal (D) can be given by the Pythagorean theorem:
\[
D^2 = W^2 + H^2
\]
From this, we can rearrange it to find the height:
\[
H = \sqrt{D^2 - W^2}
\]
Let's calculate the height for each screen:
### Screen A
- **Diagonal (D)**: 15 inches
- **Width (W)**: 13 inches
\[
H = \sqrt{15^2 - 13^2} = \sqrt{225 - 169} = \sqrt{56} \approx 7.48 \text{ inches}
\]
### Screen B
- **Diagonal (D)**: 22 inches
- **Width (W)**: 19 inches
\[
H = \sqrt{22^2 - 19^2} = \sqrt{484 - 361} = \sqrt{123} \approx 11.09 \text{ inches}
\]
### Screen C
- **Diagonal (D)**: 32 inches
- **Width (W)**: 28 inches
\[
H = \sqrt{32^2 - 28^2} = \sqrt{1024 - 784} = \sqrt{240} \approx 15.49 \text{ inches}
\]
### Screen D
- **Diagonal (D)**: 40 inches
- **Width (W)**: 35 inches
\[
H = \sqrt{40^2 - 35^2} = \sqrt{1600 - 1225} = \sqrt{375} \approx 19.36 \text{ inches}
\]
### Comparing to the Shelf Dimensions
Now we will compare the height and width of each screen with the shelf dimensions (17 inches tall and 50 inches wide):
- **Screen A**:
- Height: 7.48 inches < 17 inches ✅
- Width: 13 inches < 50 inches ✅
- **Screen B**:
- Height: 11.09 inches < 17 inches ✅
- Width: 19 inches < 50 inches ✅
- **Screen C**:
- Height: 15.49 inches < 17 inches ✅
- Width: 28 inches < 50 inches ✅
- **Screen D**:
- Height: 19.36 inches > 17 inches ❌ (does not fit)
- Width: 35 inches < 50 inches ✅
### Conclusion
Screens A, B, and C fit the shelf, but screen D does not. Among the screens that fit, screen C has the largest diagonal of 32 inches. Therefore, the largest screen Yoselin can install on her shelf is:
**Screen C (32 inches diagonal, 28 inches wide, approximately 15.49 inches tall).**
Answered by
GPT-4o mini
Let's calculate the height for each screen using the Pythagorean Theorem, given that the diagonal (D) and width (W) are provided:
### Formula
We can derive the height (H) using the formula:
\[
H = \sqrt{D^2 - W^2}
\]
### Calculations
#### **Screen A**
- **Diagonal (D)**: 15 inches
- **Width (W)**: 13 inches
\[
H = \sqrt{15^2 - 13^2} = \sqrt{225 - 169} = \sqrt{56} \approx 7.48 \text{ inches}
\]
#### **Screen B**
- **Diagonal (D)**: 22 inches
- **Width (W)**: 19 inches
\[
H = \sqrt{22^2 - 19^2} = \sqrt{484 - 361} = \sqrt{123} \approx 11.09 \text{ inches}
\]
#### **Screen C**
- **Diagonal (D)**: 32 inches
- **Width (W)**: 28 inches
\[
H = \sqrt{32^2 - 28^2} = \sqrt{1024 - 784} = \sqrt{240} \approx 15.49 \text{ inches}
\]
#### **Screen D**
- **Diagonal (D)**: 40 inches
- **Width (W)**: 35 inches
\[
H = \sqrt{40^2 - 35^2} = \sqrt{1600 - 1225} = \sqrt{375} \approx 19.36 \text{ inches}
\]
### Completed Table
Here is the completed table with the calculated heights:
| Screen | Diagonal | Width | Height |
|--------|----------|-------|---------|
| A | 15” | 13” | 7.48” |
| B | 22” | 19” | 11.09” |
| C | 32” | 28” | 15.49” |
| D | 40” | 35” | 19.36” |
This table now summarizes the diagonal, width, and calculated height for each of the screens.
### Formula
We can derive the height (H) using the formula:
\[
H = \sqrt{D^2 - W^2}
\]
### Calculations
#### **Screen A**
- **Diagonal (D)**: 15 inches
- **Width (W)**: 13 inches
\[
H = \sqrt{15^2 - 13^2} = \sqrt{225 - 169} = \sqrt{56} \approx 7.48 \text{ inches}
\]
#### **Screen B**
- **Diagonal (D)**: 22 inches
- **Width (W)**: 19 inches
\[
H = \sqrt{22^2 - 19^2} = \sqrt{484 - 361} = \sqrt{123} \approx 11.09 \text{ inches}
\]
#### **Screen C**
- **Diagonal (D)**: 32 inches
- **Width (W)**: 28 inches
\[
H = \sqrt{32^2 - 28^2} = \sqrt{1024 - 784} = \sqrt{240} \approx 15.49 \text{ inches}
\]
#### **Screen D**
- **Diagonal (D)**: 40 inches
- **Width (W)**: 35 inches
\[
H = \sqrt{40^2 - 35^2} = \sqrt{1600 - 1225} = \sqrt{375} \approx 19.36 \text{ inches}
\]
### Completed Table
Here is the completed table with the calculated heights:
| Screen | Diagonal | Width | Height |
|--------|----------|-------|---------|
| A | 15” | 13” | 7.48” |
| B | 22” | 19” | 11.09” |
| C | 32” | 28” | 15.49” |
| D | 40” | 35” | 19.36” |
This table now summarizes the diagonal, width, and calculated height for each of the screens.
Answered by
GPT-4o mini
To find the heights of each screen using the Pythagorean theorem, you can follow these step-by-step directions:
### Directions to Calculate Heights of Screens
1. **Identify Measurements**:
- For each screen, note the diagonal (D) and width (W). These values should be provided.
2. **Set Up the Formula**:
- Use the Pythagorean theorem to relate the diagonal (D), width (W), and height (H). The formula is:
\[
H = \sqrt{D^2 - W^2}
\]
3. **Square the Diagonal**:
- Calculate the square of the diagonal:
\[
D^2
\]
4. **Square the Width**:
- Calculate the square of the width:
\[
W^2
\]
5. **Subtract the Width Square from the Diagonal Square**:
- Subtract the squared width from the squared diagonal:
\[
D^2 - W^2
\]
6. **Take the Square Root**:
- Calculate the square root of the result from step 5 in order to find the height (H):
\[
H = \sqrt{D^2 - W^2}
\]
7. **Repeat for Each Screen**:
- Repeat steps 3 to 6 for each screen using their respective diagonal and width measurements.
### Example Using Screen A
1. **Identify Measurements**:
- Diagonal (D) = 15 inches, Width (W) = 13 inches
2. **Set Up the Formula**:
\[
H = \sqrt{D^2 - W^2}
\]
3. **Square the Diagonal**:
\[
D^2 = 15^2 = 225
\]
4. **Square the Width**:
\[
W^2 = 13^2 = 169
\]
5. **Subtract the Width Square from the Diagonal Square**:
\[
D^2 - W^2 = 225 - 169 = 56
\]
6. **Take the Square Root**:
\[
H = \sqrt{56} \approx 7.48 \text{ inches}
\]
By following these directions, you can calculate the height for each screen based on its diagonal and width.
### Directions to Calculate Heights of Screens
1. **Identify Measurements**:
- For each screen, note the diagonal (D) and width (W). These values should be provided.
2. **Set Up the Formula**:
- Use the Pythagorean theorem to relate the diagonal (D), width (W), and height (H). The formula is:
\[
H = \sqrt{D^2 - W^2}
\]
3. **Square the Diagonal**:
- Calculate the square of the diagonal:
\[
D^2
\]
4. **Square the Width**:
- Calculate the square of the width:
\[
W^2
\]
5. **Subtract the Width Square from the Diagonal Square**:
- Subtract the squared width from the squared diagonal:
\[
D^2 - W^2
\]
6. **Take the Square Root**:
- Calculate the square root of the result from step 5 in order to find the height (H):
\[
H = \sqrt{D^2 - W^2}
\]
7. **Repeat for Each Screen**:
- Repeat steps 3 to 6 for each screen using their respective diagonal and width measurements.
### Example Using Screen A
1. **Identify Measurements**:
- Diagonal (D) = 15 inches, Width (W) = 13 inches
2. **Set Up the Formula**:
\[
H = \sqrt{D^2 - W^2}
\]
3. **Square the Diagonal**:
\[
D^2 = 15^2 = 225
\]
4. **Square the Width**:
\[
W^2 = 13^2 = 169
\]
5. **Subtract the Width Square from the Diagonal Square**:
\[
D^2 - W^2 = 225 - 169 = 56
\]
6. **Take the Square Root**:
\[
H = \sqrt{56} \approx 7.48 \text{ inches}
\]
By following these directions, you can calculate the height for each screen based on its diagonal and width.
Answered by
GPT-4o mini
Yes, your revised instructions are clearer and guide the user more directly through the process of applying the Pythagorean theorem. Here’s a more structured and detailed version of your steps using the same approach along with a specific example.
### Directions to Calculate Heights Using the Pythagorean Theorem
1. **Start with the Pythagorean Theorem**:
\[
D^2 = W^2 + H^2
\]
where \(D\) is the diagonal, \(W\) is the width, and \(H\) is the height.
2. **Plug in the Values**:
- For each screen, substitute the known values for the diagonal \(D\) and the width \(W\) into the equation.
3. **Square the Values**:
- Calculate \(D^2\) (the square of the diagonal) and \(W^2\) (the square of the width).
4. **Rewrite the Equation**:
- Rearrange the equation to isolate \(H^2\):
\[
H^2 = D^2 - W^2
\]
5. **Subtract \(W^2\) from \(D^2\)**:
- Perform the subtraction to find \(H^2\).
6. **Take the Square Root**:
- Take the square root of both sides to solve for \(H\):
\[
H = \sqrt{H^2}
\]
7. **Calculate for Each Screen**:
- Repeat these steps for each screen by using their respective diagonal and width measurements.
### Example: Calculating Height for Screen A
1. **Start with the Pythagorean Theorem**:
\[
D^2 = W^2 + H^2
\]
2. **Plug in the Values**:
- For Screen A, \(D = 15\) inches and \(W = 13\) inches:
\[
15^2 = 13^2 + H^2
\]
3. **Square the Values**:
\[
225 = 169 + H^2
\]
4. **Rewrite the Equation**:
\[
H^2 = 225 - 169
\]
5. **Subtract \(W^2\) from \(D^2\)**:
\[
H^2 = 56
\]
6. **Take the Square Root**:
\[
H = \sqrt{56} \approx 7.48 \text{ inches}
\]
### Conclusion
By following these steps, you can systematically calculate the height of each screen. Your structured approach is easy to follow and effectively conveys the problem-solving process.
### Directions to Calculate Heights Using the Pythagorean Theorem
1. **Start with the Pythagorean Theorem**:
\[
D^2 = W^2 + H^2
\]
where \(D\) is the diagonal, \(W\) is the width, and \(H\) is the height.
2. **Plug in the Values**:
- For each screen, substitute the known values for the diagonal \(D\) and the width \(W\) into the equation.
3. **Square the Values**:
- Calculate \(D^2\) (the square of the diagonal) and \(W^2\) (the square of the width).
4. **Rewrite the Equation**:
- Rearrange the equation to isolate \(H^2\):
\[
H^2 = D^2 - W^2
\]
5. **Subtract \(W^2\) from \(D^2\)**:
- Perform the subtraction to find \(H^2\).
6. **Take the Square Root**:
- Take the square root of both sides to solve for \(H\):
\[
H = \sqrt{H^2}
\]
7. **Calculate for Each Screen**:
- Repeat these steps for each screen by using their respective diagonal and width measurements.
### Example: Calculating Height for Screen A
1. **Start with the Pythagorean Theorem**:
\[
D^2 = W^2 + H^2
\]
2. **Plug in the Values**:
- For Screen A, \(D = 15\) inches and \(W = 13\) inches:
\[
15^2 = 13^2 + H^2
\]
3. **Square the Values**:
\[
225 = 169 + H^2
\]
4. **Rewrite the Equation**:
\[
H^2 = 225 - 169
\]
5. **Subtract \(W^2\) from \(D^2\)**:
\[
H^2 = 56
\]
6. **Take the Square Root**:
\[
H = \sqrt{56} \approx 7.48 \text{ inches}
\]
### Conclusion
By following these steps, you can systematically calculate the height of each screen. Your structured approach is easy to follow and effectively conveys the problem-solving process.
Answered by
GPT-4o mini
To determine which screen is the largest that will fit on Yoselin's shelf, we will review the dimensions of each screen and compare them to the shelf's dimensions.
### Shelf Dimensions:
- **Height**: 17 inches
- **Width**: 50 inches
### Previously Calculated Dimensions for Each Screen:
| Screen | Diagonal | Width | Height | Fits? |
|--------|----------|-------|---------|-------|
| A | 15” | 13” | 7.48” | Yes |
| B | 22” | 19” | 11.09” | Yes |
| C | 32” | 28” | 15.49” | Yes |
| D | 40” | 35” | 19.36” | No |
### Evaluating Each Screen:
- **Screen A**
- Height: 7.48 inches (fits)
- Width: 13 inches (fits)
- **Screen B**
- Height: 11.09 inches (fits)
- Width: 19 inches (fits)
- **Screen C**
- Height: 15.49 inches (fits)
- Width: 28 inches (fits)
- **Screen D**
- Height: 19.36 inches (does not fit, exceeds 17 inches tall)
- Width: 35 inches (fits, but height is the limiting factor)
### Conclusion:
Among screens A, B, and C, **Screen C** is the largest screen that fits on Yoselin's shelf with a diagonal of **32 inches**, a width of **28 inches**, and a height of approximately **15.49 inches**.
Thus, the largest screen that will fit on Yoselin's shelf is **Screen C (32 inches diagonal)**.
### Shelf Dimensions:
- **Height**: 17 inches
- **Width**: 50 inches
### Previously Calculated Dimensions for Each Screen:
| Screen | Diagonal | Width | Height | Fits? |
|--------|----------|-------|---------|-------|
| A | 15” | 13” | 7.48” | Yes |
| B | 22” | 19” | 11.09” | Yes |
| C | 32” | 28” | 15.49” | Yes |
| D | 40” | 35” | 19.36” | No |
### Evaluating Each Screen:
- **Screen A**
- Height: 7.48 inches (fits)
- Width: 13 inches (fits)
- **Screen B**
- Height: 11.09 inches (fits)
- Width: 19 inches (fits)
- **Screen C**
- Height: 15.49 inches (fits)
- Width: 28 inches (fits)
- **Screen D**
- Height: 19.36 inches (does not fit, exceeds 17 inches tall)
- Width: 35 inches (fits, but height is the limiting factor)
### Conclusion:
Among screens A, B, and C, **Screen C** is the largest screen that fits on Yoselin's shelf with a diagonal of **32 inches**, a width of **28 inches**, and a height of approximately **15.49 inches**.
Thus, the largest screen that will fit on Yoselin's shelf is **Screen C (32 inches diagonal)**.
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