Asked by annoymous
What is the surface area of a square pyramid where the base edge is 1.5 cm and the slant height is 4 cm? Round your answer to the nearest hundredth.(1 point) ____cm^2
Answers
Answered by
annoymous
Question
Niko has an outdoor play tent in the form of a regular triangular pyramid, covered in fabric on all four sides. The surface area of the tent is 100 ft.2
, the base is 6 ft., and the slant height is 8 ft. What is the height of the base to the nearest tenth?(1 point)
Responses
9.3 ft.
9.3 ft.
372 ft.
372 ft.
6.5 ft.
6.5 ft.
57.3 ft.
Niko has an outdoor play tent in the form of a regular triangular pyramid, covered in fabric on all four sides. The surface area of the tent is 100 ft.2
, the base is 6 ft., and the slant height is 8 ft. What is the height of the base to the nearest tenth?(1 point)
Responses
9.3 ft.
9.3 ft.
372 ft.
372 ft.
6.5 ft.
6.5 ft.
57.3 ft.
Answered by
GPT-4o mini
To calculate the surface area of a square pyramid, we need to find both the area of the base and the area of the four triangular faces.
1. **Calculate the area of the base:**
The base of the pyramid is a square. The area \( A_{base} \) of a square is given by the formula:
\[
A_{base} = s^2
\]
where \( s \) is the length of the edge of the base. Given that \( s = 1.5 \) cm:
\[
A_{base} = (1.5 \, \text{cm})^2 = 2.25 \, \text{cm}^2
\]
2. **Calculate the area of one triangular face:**
Each triangular face has a base equal to the edge of the square base and a height equal to the slant height. The area \( A_{triangle} \) of a triangle is given by the formula:
\[
A_{triangle} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base of the triangle is 1.5 cm and the height (slant height) is 4 cm:
\[
A_{triangle} = \frac{1}{2} \times 1.5 \, \text{cm} \times 4 \, \text{cm} = \frac{1}{2} \times 6 \, \text{cm}^2 = 3 \, \text{cm}^2
\]
3. **Calculate the total area of the four triangular faces:**
Since there are four identical triangular faces, the total area \( A_{faces} \) is:
\[
A_{faces} = 4 \times A_{triangle} = 4 \times 3 \, \text{cm}^2 = 12 \, \text{cm}^2
\]
4. **Calculate the total surface area of the pyramid:**
The total surface area \( A_{total} \) of the pyramid is the sum of the area of the base and the total area of the triangular faces:
\[
A_{total} = A_{base} + A_{faces} = 2.25 \, \text{cm}^2 + 12 \, \text{cm}^2 = 14.25 \, \text{cm}^2
\]
Finally, rounding to the nearest hundredth, the surface area of the square pyramid is:
\[
\text{Surface Area} = 14.25 \, \text{cm}^2
\]
So, the final answer is:
**14.25 cm²**
1. **Calculate the area of the base:**
The base of the pyramid is a square. The area \( A_{base} \) of a square is given by the formula:
\[
A_{base} = s^2
\]
where \( s \) is the length of the edge of the base. Given that \( s = 1.5 \) cm:
\[
A_{base} = (1.5 \, \text{cm})^2 = 2.25 \, \text{cm}^2
\]
2. **Calculate the area of one triangular face:**
Each triangular face has a base equal to the edge of the square base and a height equal to the slant height. The area \( A_{triangle} \) of a triangle is given by the formula:
\[
A_{triangle} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base of the triangle is 1.5 cm and the height (slant height) is 4 cm:
\[
A_{triangle} = \frac{1}{2} \times 1.5 \, \text{cm} \times 4 \, \text{cm} = \frac{1}{2} \times 6 \, \text{cm}^2 = 3 \, \text{cm}^2
\]
3. **Calculate the total area of the four triangular faces:**
Since there are four identical triangular faces, the total area \( A_{faces} \) is:
\[
A_{faces} = 4 \times A_{triangle} = 4 \times 3 \, \text{cm}^2 = 12 \, \text{cm}^2
\]
4. **Calculate the total surface area of the pyramid:**
The total surface area \( A_{total} \) of the pyramid is the sum of the area of the base and the total area of the triangular faces:
\[
A_{total} = A_{base} + A_{faces} = 2.25 \, \text{cm}^2 + 12 \, \text{cm}^2 = 14.25 \, \text{cm}^2
\]
Finally, rounding to the nearest hundredth, the surface area of the square pyramid is:
\[
\text{Surface Area} = 14.25 \, \text{cm}^2
\]
So, the final answer is:
**14.25 cm²**
Answered by
GPT-4o mini
To find the height of the base of the triangular pyramid (tent) given the surface area, base length, and slant height, we can follow these steps:
1. **Identify the components:**
- The tent is a regular triangular pyramid.
- The surface area (SA) of the pyramid is 100 ft².
- The length of the base \( b = 6 \) ft.
- The slant height \( l = 8 \) ft.
2. **Calculate the area of the triangular base:**
For a regular triangular pyramid, the area of the base can be calculated using the formula for the area of an equilateral triangle:
\[
A_{base} = \frac{\sqrt{3}}{4} b^2
\]
Substituting the value of the base:
\[
A_{base} = \frac{\sqrt{3}}{4} (6^2) = \frac{\sqrt{3}}{4} (36) = 9\sqrt{3} \, \text{ft}^2
\]
Approximating \( \sqrt{3} \approx 1.732 \):
\[
A_{base} \approx 9 \times 1.732 \approx 15.588 \, \text{ft}^2
\]
3. **Find the area of the three triangular faces:**
The total surface area of the pyramid consists of the area of the base plus the area of the three triangular faces:
\[
SA = A_{base} + A_{faces}
\]
Therefore, the area of the faces can be expressed as:
\[
A_{faces} = SA - A_{base} = 100 \, \text{ft}^2 - 9\sqrt{3} \, \text{ft}^2
\]
Approximating \( 9\sqrt{3} \):
\[
A_{faces} \approx 100 - 15.588 \approx 84.412 \, \text{ft}^2
\]
4. **Area of one triangular face:**
The area of one triangular face can be calculated using the formula for the area of a triangle:
\[
A_{triangle} = \frac{1}{2} \times b \times l
\]
For three identical triangular faces:
\[
A_{faces} = 3 \times A_{triangle} = 3 \left( \frac{1}{2} \times 6 \, \text{ft} \times 8 \, \text{ft} \right) = 3 \left( \frac{1}{2} \times 48 \, \text{ft}^2 \right) = 3 \times 24 \, \text{ft}^2 = 72 \, \text{ft}^2
\]
5. **Verify and find height of the small triangles:**
To find the height of the triangular face, we need to find the height \( h \) of the triangular face using its area.
The area of one triangular face is:
\[
A_{triangle} = 3 \left( \frac{1}{2} \times b \times h \right) = 3 \left( \frac{1}{2} \times 6 \times h \right) = 9h
\]
We know that the total area of the triangular faces must match the \( A_{faces} \):
\[
72 = 9h \implies h = \frac{72}{9} = 8 \, \text{ft}
\]
6. **Calculate height of base triangle:**
The height of the equilateral triangle's base can be calculated using the following formula:
\[
h_{base} = \frac{\sqrt{3}}{2} b
\]
Substituting the value:
\[
h_{base} = \frac{\sqrt{3}}{2} \times 6 \approx \frac{1.732 \times 6}{2} \approx 5.196 \, \text{ft}
\]
Thus, rounding to the nearest tenth, the height of the base is approximately:
**5.2 ft** (This value isn't listed in your choices, please cross-verify data.)
If your choices suggest a reported height which seems inconsistent due to miscalculations, feel free to specify them. However, following the outlined calculations above, I can't find a match exactly in your options provided.
1. **Identify the components:**
- The tent is a regular triangular pyramid.
- The surface area (SA) of the pyramid is 100 ft².
- The length of the base \( b = 6 \) ft.
- The slant height \( l = 8 \) ft.
2. **Calculate the area of the triangular base:**
For a regular triangular pyramid, the area of the base can be calculated using the formula for the area of an equilateral triangle:
\[
A_{base} = \frac{\sqrt{3}}{4} b^2
\]
Substituting the value of the base:
\[
A_{base} = \frac{\sqrt{3}}{4} (6^2) = \frac{\sqrt{3}}{4} (36) = 9\sqrt{3} \, \text{ft}^2
\]
Approximating \( \sqrt{3} \approx 1.732 \):
\[
A_{base} \approx 9 \times 1.732 \approx 15.588 \, \text{ft}^2
\]
3. **Find the area of the three triangular faces:**
The total surface area of the pyramid consists of the area of the base plus the area of the three triangular faces:
\[
SA = A_{base} + A_{faces}
\]
Therefore, the area of the faces can be expressed as:
\[
A_{faces} = SA - A_{base} = 100 \, \text{ft}^2 - 9\sqrt{3} \, \text{ft}^2
\]
Approximating \( 9\sqrt{3} \):
\[
A_{faces} \approx 100 - 15.588 \approx 84.412 \, \text{ft}^2
\]
4. **Area of one triangular face:**
The area of one triangular face can be calculated using the formula for the area of a triangle:
\[
A_{triangle} = \frac{1}{2} \times b \times l
\]
For three identical triangular faces:
\[
A_{faces} = 3 \times A_{triangle} = 3 \left( \frac{1}{2} \times 6 \, \text{ft} \times 8 \, \text{ft} \right) = 3 \left( \frac{1}{2} \times 48 \, \text{ft}^2 \right) = 3 \times 24 \, \text{ft}^2 = 72 \, \text{ft}^2
\]
5. **Verify and find height of the small triangles:**
To find the height of the triangular face, we need to find the height \( h \) of the triangular face using its area.
The area of one triangular face is:
\[
A_{triangle} = 3 \left( \frac{1}{2} \times b \times h \right) = 3 \left( \frac{1}{2} \times 6 \times h \right) = 9h
\]
We know that the total area of the triangular faces must match the \( A_{faces} \):
\[
72 = 9h \implies h = \frac{72}{9} = 8 \, \text{ft}
\]
6. **Calculate height of base triangle:**
The height of the equilateral triangle's base can be calculated using the following formula:
\[
h_{base} = \frac{\sqrt{3}}{2} b
\]
Substituting the value:
\[
h_{base} = \frac{\sqrt{3}}{2} \times 6 \approx \frac{1.732 \times 6}{2} \approx 5.196 \, \text{ft}
\]
Thus, rounding to the nearest tenth, the height of the base is approximately:
**5.2 ft** (This value isn't listed in your choices, please cross-verify data.)
If your choices suggest a reported height which seems inconsistent due to miscalculations, feel free to specify them. However, following the outlined calculations above, I can't find a match exactly in your options provided.
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