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, we need to calculate the length of a line perpendicular to the base from the vertex to the section.

First, let's find the area of the section parallel to the base. We are given the area as 72 cm^2.

The formula for the area of a triangle is: Area = (1/2) * Base * Height.

Since the section is parallel to the base, the base of the section is equal to the base of the rectangle, which is 18 cm.

Let's substitute the values into the formula:
72 cm^2 = (1/2) * 18 cm * Height.

Now, we can solve for the height:
72 cm^2 = 9 cm * Height.
Height = 72 cm^2 / 9 cm = 8 cm.

So, the height of the section parallel to the base is 8 cm.

Since the altitude of the pyramid is given as 30 cm, we can now use the Pythagorean theorem to find the distance from the vertex to the section.

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides.

In our case, the height of the section is one side, the distance to the section is the second side, and the altitude is the hypotenuse.

Let's denote the distance to the section as "x".

Using the Pythagorean theorem, we have:
x^2 + 8^2 = 30^2.

Simplifying the equation:
x^2 + 64 = 900.
x^2 = 900 - 64.
x^2 = 836.

Taking the square root of both sides:
x = √836.

Using a calculator, we find:
x ≈ 28.93 cm.

Therefore, the distance from the vertex of the section to the base is approximately 28.93 cm.