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

Answer 1

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

Answer 2

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.

Answer 3

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.