A sphere of radius 7 m is circumscribed about a regular triangular prism. If the base edge of the prism is 1 m, find the volume of the prism (in m3). Round off your answer to four decimal places.
To find the volume of the prism, we need to know its height. However, this information is not given.
However, we can still proceed by using the relationship between the radius of the circumscribed sphere and the length of the prism.
The radius of the sphere is given to be 7 m. This radius is the distance from the center of the sphere to any of its points on the surface. Since the sphere is circumscribed about the prism, its center would be the same as the center of the prism's base.
A regular triangular prism has equilateral triangular bases. The base edge of the prism is given to be 1 m. In an equilateral triangle, all sides are equal. Therefore, each side of the equilateral triangle formed by the base of the prism measures 1 m.
Let's draw a right triangle from the center of the sphere to one of the vertices of the equilateral triangle. The hypotenuse of this right triangle would be the radius of the sphere, which is 7 m. One of the legs would be half of one side of the equilateral triangle, which is 1/2 m.
Using the Pythagorean theorem, we can solve for the height of the triangle, which would be equal to the height of the prism:
height^2 = hypotenuse^2 - leg^2
height^2 = 7^2 - (1/2)^2
height^2 = 49 - 1/4
height^2 = 195/4
height ≈ √(195/4)
height ≈ 6.2409 m (rounded to four decimal places)
Now that we have the height of the prism, we can calculate its volume. The volume of a prism is given by the formula: V = base area * height.
The base area of a regular triangular prism can be found by finding the area of one of the equilateral triangles and multiplying it by the length of the prism:
Area of an equilateral triangle = (sqrt(3)/4) * side^2
In this case, the side length is 1 m:
Area of an equilateral triangle = (sqrt(3)/4) * (1^2) = sqrt(3)/4
Now we can calculate the volume:
V = base area * height
V = (sqrt(3)/4) * 1 * 6.2409
V ≈ 1.0798 m^3 (rounded to four decimal places)
Therefore, the volume of the prism is approximately 1.0798 cubic meters.