Architecture students are given an assignment to build a rectangularly laid out house with a volume of 20,000 cubic feet.
What are some possible dimensions for this house? Use these dimensions for the activity that follows.
The roof is a triangular prism placed atop the house’s main frame. The base of the triangular prism is an isosceles triangle, and the length of the base of this triangle must be 4 feet longer than the length of the house.
An illustration shows a rectangular based pyramid stacked on top of a 3 D rectangular prism. The pyramid extends slightly past the prism on both sides. The length of the pyramid that extends past the prism is labeled 2 feet on each side.
The perpendicular height from the apex of the pyramid to the top of the rectangular prism, denoted by a right angle symbol, is 12 feet.
What are the dimensions of the house?
What is the surface area of the roof that will need to be tiled?
If each tile is 12’’ x 24’’, how many tiles will be required to cover the entire roof?
To find the dimensions of the house, let's assume the length of the house is x feet.
Given that the base of the triangular prism (the roof) is an isosceles triangle and the length of the base must be 4 feet longer than the length of the house, the length of the base of the triangular prism is (x + 4) feet.
Since the triangular prism extends slightly past the rectangular prism on both sides, this means the length of the pyramid that extends past the prism is 2 feet on each side. Therefore, the total length of the triangular prism (roof) is (x + 4 + 2 + 2) feet, which simplifies to (x + 8) feet.
The given height from the apex of the pyramid to the top of the rectangular prism is 12 feet.
Let's calculate the dimensions of the rectangular prism first:
- Length = x feet
- Width = unknown
- Height = unknown
To find the width and height of the rectangular prism, we can use the volume formula:
Volume of rectangular prism = Length * Width * Height
20,000 = x * Width * Height
Now, let's find the dimensions of the triangular prism (roof):
- Length = (x + 8) feet
- Width = unknown
- Height = 12 feet
To find the width of the triangular prism, we can use the formula for the volume of a triangular prism:
Volume of triangular prism = (1/2) * base * height * length
20,000 = (1/2) * (x + 4) * 12 * (x + 8)
(Note: The base of the triangular prism is the isosceles triangle, whose length is x + 4, and the length is x + 8)
Now, we have two equations:
1) 20,000 = x * Width * Height
2) 20,000 = (1/2) * (x + 4) * 12 * (x + 8)
Using these equations, we can solve for x and the dimensions of the house.
As for the surface area of the roof that will need to be tiled, it can be calculated by finding the area of the triangular faces of the roof.
Surface area of each triangular face = (1/2) * base * height
Surface area of each triangular face = (1/2) * (x + 4) * 12
Since there are two triangular faces, the total surface area of the roof that needs to be tiled is:
Total surface area of roof = 2 * surface area of each triangular face
Finally, to calculate the number of tiles required to cover the entire roof, we need to divide the total surface area of the roof by the area of each tile and round up to the nearest whole number.
Number of tiles required = Ceil(Total surface area of roof / Area of each tile)
1. What are the dimensions of the house?
2. What is the surface area of the roof that will need to be tiled?
3. If each tile is 12’’ x 24’’, how many tiles will be required to cover the entire roof?
To find the dimensions of the house, we need to solve the set of equations:
1) 20,000 = x * Width * Height
2) 20,000 = (1/2) * (x + 4) * 12 * (x + 8)
Unfortunately, without specific information about the width and height of the rectangular prism (house), we cannot determine the exact dimensions.
To calculate the surface area of the roof that will need to be tiled, we need to find the area of each triangular face and then sum them up:
Surface area of each triangular face = (1/2) * base * height
Surface area of each triangular face = (1/2) * (x + 4) * 12
Since there are two triangular faces, the total surface area of the roof that needs to be tiled can be calculated as:
Total surface area of roof = 2 * surface area of each triangular face
To find the number of tiles required to cover the entire roof, we need to divide the total surface area of the roof by the area of each tile. Since each tile is 12'' x 24'', the area of each tile is 12 * 24 = 288 square inches.
To convert the total surface area of the roof from square feet to square inches (to match the unit of the tile's area), we need to multiply it by 144 (since there are 144 square inches in a square foot).
After obtaining the total surface area of the roof in square inches, we can divide it by the area of each tile to find the number of tiles required. Finally, we need to round up to the nearest whole number since we cannot have a fraction of a tile.
Please provide the values for the width and height of the house for a more precise calculation.