A vase in the shape of a sqare based pyramid just fits inside a box in the shape of a square based prism. If the prism has the volume of 720 cm then what is the volume of the vase?
A pyramid (coming to a point or vertex) with a certain base has 1/3 the volume of the prism with the same base and height.
so the pyramid is (1/3)(270) = 90
can telll
To find the volume of the vase, we need to determine the volume of the square based pyramid. We know that the pyramid just fits inside the square based prism, so the base of the pyramid must have the same area as the base of the prism.
Since the base of the prism is in the shape of a square, we can calculate the area of the base by finding the side length and then squaring it. To find the side length, we can take the square root of the volume of the prism.
Let's calculate the side length of the square based prism first:
√(720) ≈ 26.83 cm
Now that we have the side length of the square based prism, we can calculate the volume of the vase. The volume of a square based pyramid can be calculated using the formula:
Volume = (1/3) * base area * height
Since the pyramid fits perfectly inside the prism, the height of the pyramid would be equal to the side length of the prism.
Height = 26.83 cm
The base area of the pyramid is the same as the base area of the prism, which can be calculated by squaring the side length.
Base area = (26.83 cm)^2 = 719.55 cm^2 (rounded to two decimal places)
Now we can calculate the volume of the vase:
Volume = (1/3) * base area * height
= (1/3) * 719.55 cm^2 * 26.83 cm
≈ 6,106.20 cm^3
Therefore, the volume of the vase is approximately 6,106.20 cm^3.
To find the volume of the vase, we need to determine the volume of the square based pyramid.
Let's start by breaking down the information provided:
1. The box is in the shape of a square-based prism, and its volume is given as 720 cm³.
2. The vase is in the shape of a square-based pyramid and fits inside the box.
Since the vase fits inside the box, we can conclude that the volume of the vase will be less than or equal to the volume of the box. We also know that the box is a prism, which means its volume can be calculated using the formula:
Volume of a prism = Base area × Height
Let's assume the side length of the base of the square-based prism is "a" cm, and the height of the prism is "h" cm.
Since the box is a square-based prism, the base area of the prism will be equal to the area of a square. Therefore, the base area is given by:
Base area = a × a = a² cm²
We also know that the volume of the square-based prism is given as 720 cm³:
Volume of the prism = Base area × Height = a² × h = 720 cm³
Now, since the vase fits inside the box, it will have the same base dimensions as the base of the prism. Therefore, the base area of the vase will also be a² cm².
The volume of the square-based pyramid can be calculated using the formula:
Volume of a pyramid = (Base area × Height) / 3
Let's denote the volume of the vase as V (cm³). Using the given information, we can set up an equation:
V = (a² × h) / 3
Since we are looking for the volume of the vase, we need to solve for V.
We have the equation V = (a² × h) / 3, and the equation a² × h = 720 cm³.
We can substitute the value of a² × h from the second equation into the first equation:
V = (720 cm³) / 3
V = 240 cm³
Therefore, the volume of the vase is 240 cm³.