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. (1 point)
To find the surface area of a triangular pyramid, we need to find the area of each of the four triangular faces and the base.
First, let's find the area of one of the triangular faces. The formula for the area of a triangle is given by:
Area = (1/2) * base * height
In the case of a triangular pyramid, the base of each triangular face is an equilateral triangle. An equilateral triangle has all sides and angles equal. Let's say that each side of the triangular face has a length of s.
The height of the triangular face can be found by using the Pythagorean theorem. Let's say that the height can be represented by h.
Using the Pythagorean theorem:
s^2 = (1/2 * s)^2 + h^2
Simplifying:
s^2 = (1/4) * s^2 + h^2
s^2 - (1/4) * s^2 = h^2
(3/4) * s^2 = h^2
h = sqrt((3/4) * s^2)
Now, let's plug in the values in the formula for the area of a triangle:
Area = (1/2) * s * sqrt((3/4) * s^2)
Simplifying further:
Area = (1/2) * s * (sqrt(3)/2) * s
Area = (sqrt(3)/4) * s^2
Since the triangular pyramid has four four triangular faces that are all the same, the total area of all the triangular faces is:
4 * (sqrt(3)/4) * s^2
= sqrt(3) * s^2
Next, let's find the area of the base. Since the base of each triangular face is an equilateral triangle, we can use the formula for the area of an equilateral triangle:
Area = (sqrt(3)/4) * s^2
Finally, to find the total surface area of the triangular pyramid, we add the area of the four triangular faces and the area of the base:
Total surface area = 4 * (sqrt(3)/4) * s^2 + (sqrt(3)/4) * s^2
= (4 * sqrt(3) + sqrt(3))/4 * s^2
= (5 * sqrt(3))/4 * s^2
So, the surface area of each triangular pyramid is (5 * sqrt(3))/4 * s^2, where s is the length of each side of the equilateral triangular base.
What is the length of a right rectangular prism with a volume of 90, width of 3, and height of 5?(1 point)
To find the length of the right rectangular prism, we can use the formula for the volume of a rectangular prism:
Volume = length * width * height
Given that the volume is 90, the width is 3, and the height is 5, we can substitute these values into the formula:
90 = length * 3 * 5
To solve for the length, we can divide both sides of the equation by (3 * 5):
length = 90 / (3 * 5)
length = 90 / 15
length = 6
Therefore, the length of the right rectangular prism is 6 units.
A right rectangular prism has an area of 20 square units and a height of 3 units. What is the volume?(1 point)
To find the volume of a right rectangular prism, we can use the formula:
Volume = length * width * height
Given that the area is 20, the height is 3, and we need to find the volume, we need to find the length and width.
The formula for the area of a rectangle is given by:
Area = length * width
Given that the area is 20, we can set up the equation:
20 = length * width
To find the volume, we need to solve for both the length and the width.
Let's consider different possible values for the length and width:
- If the length is 1 and the width is 20, the area would be 1 * 20 = 20, which matches the given area. However, this would result in a very thin prism, so this is unlikely.
- If the length is 2 and the width is 10, the area would be 2 * 10 = 20, which again matches the given area. This is a plausible scenario.
- If the length is 5 and the width is 4, the area would be 5 * 4 = 20, which once again matches the given area. This is another plausible scenario.
Since there could be multiple valid combinations of length and width that result in the given area, we'll consider both scenarios to find the possible volume values.
1) If the length is 2 and the width is 10:
Volume = length * width * height
Volume = 2 * 10 * 3
Volume = 60 units cubed
2) If the length is 5 and the width is 4:
Volume = length * width * height
Volume = 5 * 4 * 3
Volume = 60 units cubed
Therefore, the volume of the right rectangular prism could be 60 units cubed.
A gift box is being constructed in the shape of a right rectangular prism. The base is 15 inches squared and the height is 5 inches. What will the volume of the gift box be?(1 point) in.3
To find the volume of the gift box, we can multiply the area of the base by the height of the prism.
Given that the base has an area of 15 square inches and the height is 5 inches, we can substitute these values into the formula for volume:
Volume = area of base * height
Volume = 15 square inches * 5 inches
To find the product, we can simply multiply the numbers:
Volume = 75 cubic inches
Therefore, the volume of the gift box will be 75 cubic inches.
What is the volume of a rectangular pyramid with a length of 32 cm, a width of 25 cm, and an altitude of 41 cm?(1 point) Responses 32,800 cm3 32,800 cm cubed 98,400 cm3 98,400 cm cubed 32.67 cm3 32.67 cm cubed 10,933.33 cm3
To find the volume of a rectangular pyramid, we can use the formula:
Volume = (1/3) * base area * altitude
Given that the length of the rectangular pyramid is 32 cm, the width is 25 cm, and the altitude is 41 cm, we can calculate the volume using these values.
First, let's find the base area:
Base area = length * width
Base area = 32 cm * 25 cm
Base area = 800 cm²
Next, let's substitute the base area and altitude into the volume formula:
Volume = (1/3) * 800 cm² * 41 cm
Volume = 32,800 cm³
Therefore, the volume of the rectangular pyramid is 32,800 cm³.