Let's address each question one by one:
1. Surface area of a rectangular pyramid
For a rectangular pyramid, the surface area (SA) can be calculated using the formula:
\[ SA = \text{Base Area} + \text{Lateral Surface Area} \]
-
Base Area \(= L \times W\)
\(= 15, m \times 12, m = 180, m^2\) -
Lateral Surface Area: The lateral surface area consists of two triangular sides with base \(L\) and height \(h_l\) and two with base \(W\) and height \(h_w\). First, we need to find the slant heights using the Pythagorean theorem.
For the triangles with base L: \[ h_l = \sqrt{\left(\frac{L}{2}\right)^2 + h^2} = \sqrt{\left(\frac{15}{2}\right)^2 + 10^2} = \sqrt{56.25 + 100} = \sqrt{156.25} = 12.5, m \]
For the triangles with base W: \[ h_w = \sqrt{\left(\frac{W}{2}\right)^2 + h^2} = \sqrt{\left(\frac{12}{2}\right)^2 + 10^2} = \sqrt{36 + 100} = \sqrt{136} \approx 11.62, m \]
The area of the triangles is: \[ \text{Area of two triangles with base L} = 2 \left(\frac{1}{2} \cdot L \cdot h_l\right) = L \cdot h_l = 15 \cdot 12.5 = 187.5, m^2 \] \[ \text{Area of two triangles with base W} = 2 \left(\frac{1}{2} \cdot W \cdot h_w\right) = W \cdot h_w = 12 \cdot 11.62 \approx 139.44, m^2 \]
Therefore, the total lateral area is: \[ 187.5 + 139.44 \approx 326.94, m^2 \]
The surface area is: \[ SA = 180 + 326.94 \approx 506.94, m^2 \]
Answer: 506.94 m² (rounded to the nearest hundredth)
2. Surface area of a square pyramid
For a square pyramid, the surface area (SA) is calculated as follows:
-
Base Area \(= a^2\)
\(= (1.5, cm)^2 = 2.25, cm^2\) -
Lateral Surface Area: The lateral surface area is given by the formula: \[ \text{Lateral Area} = 2 \cdot a \cdot h_l \] where \(h_l\) is the slant height. \[ \text{Lateral Area} = 2 \cdot 1.5 \cdot 4 = 12, cm^2 \]
-
Total Surface Area: \[ SA = \text{Base Area} + \text{Lateral Area} = 2.25 + 12 = 14.25, cm^2 \]
Answer: 14.25 cm² (rounded to the nearest hundredth)
3. Height of the base of a triangular pyramid (tent)
The surface area of the tent covers all four triangular faces. The formula for the total surface area \(SA\) of a triangular pyramid is: \[ SA = \text{base area} + \text{lateral area} \]
- The base \(= \frac{\sqrt{3}}{4}s^2\) where \(s\) is the side length, but since we already know the base area: \[ \text{base area} = \frac{1}{2} \cdot b \cdot h, \] where \(b=6\) and we need to find height \(h\) using slant height.
Using \(S = h(h=6ft)\): \[ 100 = \frac{1}{2} \cdot 6 \cdot h + 3 \cdot 6 \] This is a complicated approach, thus must relate height and slant height. Using formulas, we also use the triangular area.
After calculating given parameters, we get: Height = 5.196 while slant height remains.
Answer: 5.2 ft (rounded to the nearest tenth)
4. Length of a right rectangular prism
Using the volume formula \(V = L \cdot W \cdot H\), we can solve for length \(L\):
\[ V = 90, \quad W = 3, \quad H = 5 \] Thus, \[ 90 = L \cdot 3 \cdot 5\ 90 = 15L \implies L = \frac{90}{15} = 6 \]
Answer: 6
5. Volume of a right rectangular prism
Using the volume formula \(V = A \cdot H\):
Given area \(A = 20\) square units, and height \(H = 3\) units: \[ V = 20 \cdot 3 = 60 \]
Answer: 60 cubic units
6. Volume of the gift box (right rectangular prism)
Given the base area and height, we can calculate the volume \(V = A \cdot H\):
\[ A = 15) in², \(H = 5\) in: [ V = 15 \cdot 5 = 75)
Answer: 75 cubic inches
7. Volume of a rectangular pyramid
The volume formula for a rectangular pyramid is: [ V = \frac{1}{3} \cdot L \cdot W \cdot H \] Substituting: \[ L = 32, cm, W = 25, cm, H = 41, cm\ V = \frac{1}{3} \cdot 32 \cdot 25 \cdot 41 = \frac{1}{3} \cdot 32800 = \frac{32800}{3} \approx 10933.33 \]
Answer: 10933.33 cubic cm
Feel free to ask further questions or clarifications!
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