Question

1. geometry the surface area of the square pyramid shown is 84 square inches. what is the value of x? the chain can support the light cover? explain. 2 ft 6 in. x in. base

Explanation:

Step1: Find the area of the base

The base is a square with side length \( 6 \) inches. The area of a square is \( s^2 \), so the area of the base is \( 6^2 = 36 \) square inches.

Step2: Find the area of the four triangular faces

The total surface area of the square pyramid is \( 84 \) square inches. Subtract the area of the base from the total surface area to get the area of the four triangular faces: \( 84 - 36 = 48 \) square inches.

Step3: Find the area of one triangular face

Since there are four identical triangular faces, the area of one triangular face is \( \frac{48}{4} = 12 \) square inches.

Step4: Use the formula for the area of a triangle to find \( x \)

The formula for the area of a triangle is \( \frac{1}{2} \times \text{base} \times \text{height} \). For each triangular face, the base is \( 6 \) inches (the side length of the square base) and the height is \( x \) inches. So we have \( \frac{1}{2} \times 6 \times x = 12 \). Simplify this equation: \( 3x = 12 \). Then solve for \( x \) by dividing both sides by \( 3 \): \( x = \frac{12}{3} = 4 \).

Answer:

\( x = 4 \)