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?
To find the height of the base, we need to find the height of the pyramid.
The surface area of a triangular pyramid is given by the formula:
Surface Area = (Base Area) + (Lateral Area)
The base area of the pyramid is the area of a triangle, which is given by the formula:
Base Area = (1/2) * Base * Height
Substituting the given values, we have:
100 ft^2 = (1/2) * 6 ft * Height + (Lateral Area)
The lateral area of a triangular pyramid is given by the formula:
Lateral Area = (1/2) * Perimeter * Slant Height
The perimeter of the base triangle is 3 times the length of one side, so in this case it is 3 * 6 ft = 18 ft.
Substituting the given values, we have:
100 ft^2 = (1/2) * 6 ft * Height + (1/2) * 18 ft * 8 ft
100 ft^2 = 3 ft * Height + 72 ft^2
Subtracting 72 ft^2 from both sides, we have:
28 ft^2 = 3 ft * Height
Dividing both sides by 3 ft, we have:
Height = 28 ft^2 / 3 ft
Height ≈ 9.3 ft
Therefore, the height of the base is approximately 9.3 ft.
A metalworker is creating decorative points in the form of triangular pyramids to go on top of a fence she is building. She wants to know the surface area of each pyramid. Use the surface area formula to find the surface area of each pyramid based on the figure.
Apologies, but you haven't provided a figure or any specific dimensions for the pyramid. Could you please provide more information or an image of the pyramid so that I can assist you in calculating the surface area?
The right side is pointing to a line going down to the bottom, says 3.5 cm, another at the bottom say 3.9 and another at the bottom saying 4.5 cm
To find the surface area of the triangular pyramid, we need to know the slant height and the dimensions of the base.
However, in your description, you have mentioned three side lengths (3.5 cm, 3.9 cm, and 4.5 cm), which does not provide enough information to determine the surface area accurately.
To calculate the surface area of the triangular pyramid, we need either the height from the apex to the base or the slant height from the apex to a point on the base.
Please provide additional information or specify the dimensions so that I can assist you further.
The height is 3.5 cm.
Thank you for providing the height of the triangular pyramid. With this information, we can calculate the surface area.
To find the surface area of a triangular pyramid, we need to calculate the area of the base and the area of the lateral faces.
First, let's calculate the area of the base. The base of the pyramid is a triangle, and the formula to calculate the area of a triangle is (1/2) * base * height. Given the dimensions you provided, the base of the triangle is 3.9 cm, and the height is 4.5 cm. Therefore, the area of the base is (1/2) * 3.9 cm * 4.5 cm = 8.775 cm².
Next, let's calculate the area of the lateral faces. Each lateral face of the triangular pyramid is a triangle, and the formula to calculate the area of a triangle is (1/2) * base * height. The base of each lateral face is the side length of the triangle, which is 3.5 cm, and the height is the height of the pyramid, which is also 3.5 cm. Therefore, the area of each lateral face is (1/2) * 3.5 cm * 3.5 cm = 6.125 cm².
Since the triangular pyramid has four lateral faces, the total area of the lateral faces is 4 * 6.125 cm² = 24.5 cm².
To find the total surface area, we add the area of the base and the area of the lateral faces: 8.775 cm² + 24.5 cm² = 33.275 cm².
Therefore, the surface area of the triangular pyramid is 33.275 cm².
What is the length of a right rectangular prism with a volume of 90, width of 3, and height of 5?
To find the length of the right rectangular prism, we can use the formula for volume:
Volume = Length * Width * Height
In this case, the volume is given as 90, the width is 3, and the height is 5. We can plug these values into the formula to solve for the length:
90 = Length * 3 * 5
Divide both sides of the equation by (3 * 5):
90 / (3 * 5) = Length
Simplify the equation:
90 / 15 = Length
6 = Length
Therefore, the length of the right rectangular prism is 6.