Question

the tower below is made up of a square pyramid on top of a rectangular prism. what is the total surface area of the tower? 6 centimeters (pyramid), 16 centimeters (prism), 7 centimeters (base). options: 539 square centimeters, 532 square centimeters, 665 square centimeters, 581 square centimeters

Explanation:

Step1: Surface Area of Rectangular Prism (excluding top face)

The rectangular prism has length \( l = 7 \) cm, width \( w = 7 \) cm (since the base of the pyramid is square, so the prism's top face is square with side 7), height \( h = 16 \) cm.
The surface area of a rectangular prism is \( 2(lw + lh + wh) \), but we exclude the top face (since it's covered by the pyramid). So:
\( lw + 2lh + 2wh = 7\times7 + 2\times7\times16 + 2\times7\times16 \)
\( = 49 + 224 + 224 = 497 \)

Step2: Lateral Surface Area of Square Pyramid

The square pyramid has base side \( s = 7 \) cm, slant height \( l = 6 \) cm.
Lateral surface area of a square pyramid is \( 4\times\frac{1}{2}\times s\times l \)
\( = 2\times7\times6 = 84 \)

Step3: Total Surface Area of Tower

Add the surface area of the prism (excluding top) and the lateral surface area of the pyramid:
\( 497 + 84 = 581 \)

Answer:

581 square centimeters