the altitude of the pyramid is 30cm and its base is enclosed by a rectangle whose dimension are 12cm by 18cm respectively. what is the distance from the vertex of a section parallel to the base whose area is 72cm^2?
To find the distance from the vertex of a section parallel to the base whose area is 72cm^2, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
In this case, the triangle represents the section parallel to the base of the pyramid, and we know that the area is 72cm^2. The base and height of this triangle are unknown.
Let's use the given information to find these values. We know that the altitude (height) of the pyramid is 30cm, and the base is enclosed by a rectangle with dimensions 12cm by 18cm.
Since the base of the pyramid is enclosed by a rectangle, the base of the section parallel to the base would also have these dimensions. So, the base of the triangle is 18cm.
Now we can rearrange the formula for the area of a triangle to solve for the height:
Area = (1/2) * base * height
72 cm^2 = (1/2) * 18 cm * height
Simplifying this equation:
72 cm^2 = 9 cm * height
Divide both sides by 9 cm:
8 cm = height
Therefore, the height of the triangle (section parallel to the base) is 8 cm.
To find the distance from the vertex of the section to the base of the pyramid, we will use the Pythagorean theorem.
The distance from the vertex to the base is the hypotenuse of a right triangle, with the altitude of the pyramid (30 cm) as one leg, and the height of the triangle (8 cm) as the other leg.
Using the Pythagorean theorem:
distance^2 = altitude^2 + height^2
distance^2 = 30 cm^2 + 8 cm^2
distance^2 = 900 cm^2 + 64 cm^2
distance^2 = 964 cm^2
Taking the square root of both sides:
distance = √964 cm
Therefore, the distance from the vertex of the section parallel to the base, whose area is 72cm^2, is approximately 31 cm.