We are given an open-ended square pyramid with a slant height of $60$ cm and a square base with an edge of $30$ cm. We want to find the amount of cardboard needed to construct the pyramid. This is equivalent to finding the surface area of the four triangular faces of the pyramid and the area of the square base.

1. Problem Description

We are given an open-ended square pyramid with a slant height of $60$ cm and a square base with an edge of $30$ cm. We want to find the amount of cardboard needed to construct the pyramid. This is equivalent to finding the surface area of the four triangular faces of the pyramid and the area of the square base.

2. Solution Steps

The area of the square base is the side length squared.

$A_{base} = side^2 = 30^2 = 900$ cm$^2$.

The area of one triangular face is given by half the base times the height, where the height is the slant height of the pyramid.

$A_{triangle} = \frac{1}{2} \times base \times height = \frac{1}{2} \times 30 \times 60 = 900$ cm$^2$.

Since there are four triangular faces, the total area of the triangular faces is

$4 \times A_{triangle} = 4 \times 900 = 3600$ cm$^2$.

The total area of cardboard needed is the sum of the area of the base and the area of the four triangular faces.

$A_{total} = A_{base} + 4 \times A_{triangle} = 900 + 3600 = 4500$ cm$^2$.

3. Final Answer

The amount of cardboard needed to construct one pyramid is $4500$ cm$^2$.