story:
Diego is building an architectural rendering of a house. For a portion of the roof, Diego cuts and folds a net for a triangular prism. the base of the prism is an isoceles triangle. the triangle has a base of 24 inches and a height of 5 inches. the distance between the triangular bases is 20 inches.
Based on this story, answer the following answer:
a.)
make an isometric sketch of the folded prism.
b.)
make a sketch of the net of the prism
c.) find the volume of the prism.
D.) find the surface area.
a.) To make an isometric sketch of the folded prism, we first need to understand what an isometric sketch is. An isometric sketch is a 3D representation of an object where all three dimensions - length, width, and height - are represented equally and at a 30-degree angle from each other.
To make the isometric sketch of the folded prism, we start by drawing a vertical line for the height of the prism. Then, we draw two diagonal lines originating from the top and bottom of the vertical line, representing the sides of the prism. These diagonal lines should be at 30-degree angles from the vertical line. Finally, we connect the top and bottom of the diagonal lines with two horizontal lines, representing the triangular bases of the prism. This completes the isometric sketch of the folded prism.
b.) To make a sketch of the net of the prism, we need to understand what a net is. A net is a 2D representation of a 3D object that shows how the object can be folded to form its shape.
To make the net of the prism, we start by drawing the net of the triangular base. We draw an isosceles triangle with a base of 24 inches and a height of 5 inches. Then, we draw two vertical lines extending from the two non-base vertices of the triangle, each with a length of 20 inches. These two vertical lines should be parallel to each other. Next, we connect the top ends of the vertical lines with a horizontal line, and the bottom ends of the vertical lines with another horizontal line. This completes the net of the prism.
c.) To find the volume of the prism, we can use the formula: Volume = Base Area x Height. Since the triangular base of the prism is an isosceles triangle, we can use the formula for the area of an isosceles triangle: Area = (1/2) x Base x Height.
In this case, the base of the isosceles triangle is 24 inches and the height is 5 inches. Plugging these values into the area formula, we get:
Area = (1/2) x 24 x 5 = 60 square inches.
Now, to find the volume, we multiply the base area by the height of the prism. In this case, the height is the distance between the triangular bases, which is given as 20 inches.
Volume = Base Area x Height = 60 square inches x 20 inches = 1200 cubic inches.
So, the volume of the prism is 1200 cubic inches.
d.) To find the surface area of the prism, we need to consider the area of each face. Since the prism has two triangular bases and three rectangular faces, we can calculate the surface area by adding up the areas of all these faces.
The area of each triangular base can be calculated using the same formula for the area of an isosceles triangle: Area = (1/2) x Base x Height. In this case, the base of the isosceles triangle is 24 inches and the height is 5 inches, so the area of each triangular base is (1/2) x 24 x 5 = 60 square inches.
The area of each rectangular face can be calculated using the formula: Area = Length x Width. In this case, the length is the distance between the triangular bases, which is 20 inches, and the width is the height of the prism, which is also 20 inches. So, the area of each rectangular face is 20 inches x 20 inches = 400 square inches.
Now, we can find the total surface area by adding up the areas of all the faces. Since there are two triangular bases and three rectangular faces, the total surface area is: 2 x 60 square inches (for the triangular bases) + 3 x 400 square inches (for the rectangular faces) = 120 square inches + 1200 square inches = 1320 square inches.
So, the surface area of the prism is 1320 square inches.