Question
25 ft, 40 ft, H
Apply the Pythagorean Theorem to determine the height of the square pyramid, h. Round the answer to the nearest tenth.
(1 point)
• 15.0 ft.
• 47.2 ft.
• 32.0 ft.
• 31.2 ft.
Apply the Pythagorean Theorem to determine the height of the square pyramid, h. Round the answer to the nearest tenth.
(1 point)
• 15.0 ft.
• 47.2 ft.
• 32.0 ft.
• 31.2 ft.
Answers
Answer
U have to pick one of the answers
Answer
The length of the prism is 32 cm, the width is 24 cm, the height is 44 cm, and the length of the diagonal of the base, segment BH, is 40 cm. Find the length of the diagonal of the rectangular prism, segment BE. Round the answer to the nearest tenth.
(1 point)
• 46.6 cm
• 51.2 cm
• 59.5 cm
• 54.4 ст
(1 point)
• 46.6 cm
• 51.2 cm
• 59.5 cm
• 54.4 ст
Answer
Chem Tech makes paper cones used to filter liquids in lab experiments. To make the paper cones, the company starts with the net of the shape, where the slant height is needed. The company wants the height of the cones to be 6 cm. If they make the diameter 13 cm, what must the slant height be? Use the Pythagorean Theorem and round the answer to the nearest tenth.
(1 point)
• 14.3 cm
• 2.5 cm
8.8 cm
• 11.5 cm
(1 point)
• 14.3 cm
• 2.5 cm
8.8 cm
• 11.5 cm
Answer
A circus tent is in the shape of a square pyramid. Each edge of the square base is 20 m and its slant height is 13.5 m. Find the height of the tent. Round the answer to the nearest tenth. (1 point)
24.1 cm
• 14.8 ст
9.1 m
• 16.8 m
24.1 cm
• 14.8 ст
9.1 m
• 16.8 m
Answer
Melissa has a gerbil that lives in a cage in the shape of a rectangular prism.
She wants to put a ramp in the cage to give her gerbil exercise. The cage has a length of 70 cm, a width of 35 cm, and a height of 50 cm. How long does the ramp need to be to fit diagonally in the cage? Round the answer to the nearest tenth. (1 point)
• 60.2 cm
• 92.9 ст
• 116.3 cm
• 78.6 cm
She wants to put a ramp in the cage to give her gerbil exercise. The cage has a length of 70 cm, a width of 35 cm, and a height of 50 cm. How long does the ramp need to be to fit diagonally in the cage? Round the answer to the nearest tenth. (1 point)
• 60.2 cm
• 92.9 ст
• 116.3 cm
• 78.6 cm
Answered by
GPT-4o mini
To find the height \( H \) of the square pyramid using the Pythagorean Theorem, we can visualize the pyramid. The base is a square with side length of 25 ft, and we can use the midpoint of the base and the apex (top point of the pyramid) to form a right triangle.
1. The half-length of the base (from the center to a vertex) can be calculated as:
\[
\text{Half the base length} = \frac{25 \text{ ft}}{2} = 12.5 \text{ ft}
\]
2. The horizontal distance from the center of the base to a vertex is 12.5 ft, and the slant height (hypotenuse of the triangle formed) is 40 ft.
3. Using the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
where:
- \( a \) is the height of the pyramid \( H \),
- \( b \) is the half-length of the base (12.5 ft),
- \( c \) is the slant height (40 ft).
Plugging in the values gives us:
\[
H^2 + (12.5)^2 = (40)^2
\]
4. Calculating \( (12.5)^2 \) and \( (40)^2 \):
\[
H^2 + 156.25 = 1600
\]
5. Now, solve for \( H^2 \):
\[
H^2 = 1600 - 156.25
\]
\[
H^2 = 1443.75
\]
6. Taking the square root to find \( H \):
\[
H = \sqrt{1443.75} \approx 38.0 \text{ ft}
\]
Since the answers provided (15.0 ft, 47.2 ft, 32.0 ft, 31.2 ft) do not include 38.0 ft, it appears there may be a misunderstanding regarding the parameters of the pyramid or an error in the problem statement. Please check the information or clarify additional measurements.
1. The half-length of the base (from the center to a vertex) can be calculated as:
\[
\text{Half the base length} = \frac{25 \text{ ft}}{2} = 12.5 \text{ ft}
\]
2. The horizontal distance from the center of the base to a vertex is 12.5 ft, and the slant height (hypotenuse of the triangle formed) is 40 ft.
3. Using the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
where:
- \( a \) is the height of the pyramid \( H \),
- \( b \) is the half-length of the base (12.5 ft),
- \( c \) is the slant height (40 ft).
Plugging in the values gives us:
\[
H^2 + (12.5)^2 = (40)^2
\]
4. Calculating \( (12.5)^2 \) and \( (40)^2 \):
\[
H^2 + 156.25 = 1600
\]
5. Now, solve for \( H^2 \):
\[
H^2 = 1600 - 156.25
\]
\[
H^2 = 1443.75
\]
6. Taking the square root to find \( H \):
\[
H = \sqrt{1443.75} \approx 38.0 \text{ ft}
\]
Since the answers provided (15.0 ft, 47.2 ft, 32.0 ft, 31.2 ft) do not include 38.0 ft, it appears there may be a misunderstanding regarding the parameters of the pyramid or an error in the problem statement. Please check the information or clarify additional measurements.
Answered by
GPT-4o mini
Apologies for any confusion earlier. Based on the calculations we performed, we arrived at a height (H) of approximately 38.0 ft, but since that is not one of the options provided, we will have to evaluate the options given.
The options you provided are:
- 15.0 ft
- 47.2 ft
- 32.0 ft
- 31.2 ft
From the calculations, the closest reasonable answer to a height derived from the given measurements is **32.0 ft**.
Thus, the option to select is **32.0 ft**.
The options you provided are:
- 15.0 ft
- 47.2 ft
- 32.0 ft
- 31.2 ft
From the calculations, the closest reasonable answer to a height derived from the given measurements is **32.0 ft**.
Thus, the option to select is **32.0 ft**.
Answered by
GPT-4o mini
To find the length of the diagonal of the rectangular prism (segment BE), we can use the 3D version of the Pythagorean theorem. The diagonal of a rectangular prism can be calculated with the formula:
\[
d = \sqrt{l^2 + w^2 + h^2}
\]
where:
- \( l \) is the length,
- \( w \) is the width,
- \( h \) is the height.
Given the dimensions of the prism:
- Length \( l = 32 \, \text{cm} \)
- Width \( w = 24 \, \text{cm} \)
- Height \( h = 44 \, \text{cm} \)
Now, we can plug in these values into the formula:
\[
d = \sqrt{(32)^2 + (24)^2 + (44)^2}
\]
Calculating each term:
1. \( (32)^2 = 1024 \)
2. \( (24)^2 = 576 \)
3. \( (44)^2 = 1936 \)
Now, add these numbers together:
\[
d = \sqrt{1024 + 576 + 1936}
\]
\[
d = \sqrt{3536}
\]
Now, calculate the square root:
\[
d \approx 59.5 \, \text{cm}
\]
Thus, rounding to the nearest tenth, the length of the diagonal of the rectangular prism (segment BE) is approximately **59.5 cm**.
So the answer is **59.5 cm**.
\[
d = \sqrt{l^2 + w^2 + h^2}
\]
where:
- \( l \) is the length,
- \( w \) is the width,
- \( h \) is the height.
Given the dimensions of the prism:
- Length \( l = 32 \, \text{cm} \)
- Width \( w = 24 \, \text{cm} \)
- Height \( h = 44 \, \text{cm} \)
Now, we can plug in these values into the formula:
\[
d = \sqrt{(32)^2 + (24)^2 + (44)^2}
\]
Calculating each term:
1. \( (32)^2 = 1024 \)
2. \( (24)^2 = 576 \)
3. \( (44)^2 = 1936 \)
Now, add these numbers together:
\[
d = \sqrt{1024 + 576 + 1936}
\]
\[
d = \sqrt{3536}
\]
Now, calculate the square root:
\[
d \approx 59.5 \, \text{cm}
\]
Thus, rounding to the nearest tenth, the length of the diagonal of the rectangular prism (segment BE) is approximately **59.5 cm**.
So the answer is **59.5 cm**.
Answered by
GPT-4o mini
To find the slant height of the paper cone, we can use the Pythagorean Theorem. A cone forms a right triangle with its height, the radius of its base, and the slant height.
Here are the steps to find the slant height:
1. **Find the radius of the base:**
The diameter of the cone is given as 13 cm, so the radius \( r \) is:
\[
r = \frac{13 \, \text{cm}}{2} = 6.5 \, \text{cm}
\]
2. **Identify the height of the cone:**
The height \( h \) of the cone is given as 6 cm.
3. **Apply the Pythagorean Theorem:**
We can use the formula:
\[
s^2 = r^2 + h^2
\]
where \( s \) is the slant height.
4. **Substituting the values:**
\[
s^2 = (6.5)^2 + (6)^2
\]
Calculating each term:
\[
(6.5)^2 = 42.25
\]
\[
(6)^2 = 36
\]
5. **Adding these values together:**
\[
s^2 = 42.25 + 36 = 78.25
\]
6. **Finding the slant height \( s \):**
\[
s = \sqrt{78.25} \approx 8.83 \, \text{cm}
\]
Rounding to the nearest tenth:
\[
s \approx 8.8 \, \text{cm}
\]
Thus, the required slant height is **8.8 cm**.
Here are the steps to find the slant height:
1. **Find the radius of the base:**
The diameter of the cone is given as 13 cm, so the radius \( r \) is:
\[
r = \frac{13 \, \text{cm}}{2} = 6.5 \, \text{cm}
\]
2. **Identify the height of the cone:**
The height \( h \) of the cone is given as 6 cm.
3. **Apply the Pythagorean Theorem:**
We can use the formula:
\[
s^2 = r^2 + h^2
\]
where \( s \) is the slant height.
4. **Substituting the values:**
\[
s^2 = (6.5)^2 + (6)^2
\]
Calculating each term:
\[
(6.5)^2 = 42.25
\]
\[
(6)^2 = 36
\]
5. **Adding these values together:**
\[
s^2 = 42.25 + 36 = 78.25
\]
6. **Finding the slant height \( s \):**
\[
s = \sqrt{78.25} \approx 8.83 \, \text{cm}
\]
Rounding to the nearest tenth:
\[
s \approx 8.8 \, \text{cm}
\]
Thus, the required slant height is **8.8 cm**.
Answered by
GPT-4o mini
To find the height of the square pyramid (the circus tent in this case), we can use the Pythagorean theorem.
Given:
- The length of each edge of the square base (side length \( s \)) is 20 m, so the half-length of the base (from the center to a vertex) is:
\[
r = \frac{s}{2} = \frac{20}{2} = 10 \, \text{m}
\]
- The slant height \( l \) is given as 13.5 m.
We want to find the height \( h \) of the pyramid. In the right triangle formed by the height, half the base, and the slant height, we can use the Pythagorean theorem:
\[
l^2 = h^2 + r^2
\]
Substituting the known values:
\[
(13.5)^2 = h^2 + (10)^2
\]
Calculating the squares:
\[
(13.5)^2 = 182.25
\]
\[
(10)^2 = 100
\]
Now substituting these values into the equation:
\[
182.25 = h^2 + 100
\]
Subtracting 100 from both sides:
\[
h^2 = 182.25 - 100
\]
\[
h^2 = 82.25
\]
Taking the square root to solve for \( h \):
\[
h = \sqrt{82.25} \approx 9.1 \, \text{m}
\]
Rounding to the nearest tenth, we find that the height of the tent is **9.1 m**.
So the answer is **9.1 m**.
Given:
- The length of each edge of the square base (side length \( s \)) is 20 m, so the half-length of the base (from the center to a vertex) is:
\[
r = \frac{s}{2} = \frac{20}{2} = 10 \, \text{m}
\]
- The slant height \( l \) is given as 13.5 m.
We want to find the height \( h \) of the pyramid. In the right triangle formed by the height, half the base, and the slant height, we can use the Pythagorean theorem:
\[
l^2 = h^2 + r^2
\]
Substituting the known values:
\[
(13.5)^2 = h^2 + (10)^2
\]
Calculating the squares:
\[
(13.5)^2 = 182.25
\]
\[
(10)^2 = 100
\]
Now substituting these values into the equation:
\[
182.25 = h^2 + 100
\]
Subtracting 100 from both sides:
\[
h^2 = 182.25 - 100
\]
\[
h^2 = 82.25
\]
Taking the square root to solve for \( h \):
\[
h = \sqrt{82.25} \approx 9.1 \, \text{m}
\]
Rounding to the nearest tenth, we find that the height of the tent is **9.1 m**.
So the answer is **9.1 m**.
Answered by
GPT-4o mini
To find the length of a ramp that fits diagonally in the cage (which is in the shape of a rectangular prism), we can use the three-dimensional Pythagorean theorem:
\[
d = \sqrt{l^2 + w^2 + h^2}
\]
where:
- \( d \) is the diagonal length,
- \( l \) is the length (70 cm),
- \( w \) is the width (35 cm),
- \( h \) is the height (50 cm).
### Step 1: Calculate each squared dimension
1. \( l^2 = (70)^2 = 4900 \)
2. \( w^2 = (35)^2 = 1225 \)
3. \( h^2 = (50)^2 = 2500 \)
### Step 2: Sum the squared dimensions
\[
d^2 = l^2 + w^2 + h^2 = 4900 + 1225 + 2500
\]
\[
d^2 = 8625
\]
### Step 3: Take the square root to find \( d \)
\[
d = \sqrt{8625} \approx 93.0 \, \text{cm}
\]
### Step 4: Round to the nearest tenth
The approximate length of the ramp needed to fit diagonally in the cage is **93.0 cm**.
Among the options provided, the closest rounded measurement is **92.9 cm**.
So the answer is **92.9 cm**.
\[
d = \sqrt{l^2 + w^2 + h^2}
\]
where:
- \( d \) is the diagonal length,
- \( l \) is the length (70 cm),
- \( w \) is the width (35 cm),
- \( h \) is the height (50 cm).
### Step 1: Calculate each squared dimension
1. \( l^2 = (70)^2 = 4900 \)
2. \( w^2 = (35)^2 = 1225 \)
3. \( h^2 = (50)^2 = 2500 \)
### Step 2: Sum the squared dimensions
\[
d^2 = l^2 + w^2 + h^2 = 4900 + 1225 + 2500
\]
\[
d^2 = 8625
\]
### Step 3: Take the square root to find \( d \)
\[
d = \sqrt{8625} \approx 93.0 \, \text{cm}
\]
### Step 4: Round to the nearest tenth
The approximate length of the ramp needed to fit diagonally in the cage is **93.0 cm**.
Among the options provided, the closest rounded measurement is **92.9 cm**.
So the answer is **92.9 cm**.