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? image: square pyramid (6 centimeters) on rectangular prism (16 centimeters height, 7 centimeters base) options: 539 square centimeters, 665 square centimeters, 581 square centimeters, 532 square centimeters

Explanation:

Step1: Analyze the rectangular prism

The rectangular prism has length \( l = 7 \) cm, width \( w = 7 \) cm (since the base of the pyramid is square, so the prism's base is square), and height \( h = 16 \) cm.
The surface area of a rectangular prism is \( 2(lw + lh + wh) \), but we need to subtract the area of the top face (since it's covered by the pyramid) and add the lateral surface area of the pyramid.
First, calculate the lateral surface area of the prism (excluding the top face):
\( lw + 2(lh + wh) = 7\times7 + 2(7\times16 + 7\times16) \)
\( = 49 + 2(112 + 112) \)
\( = 49 + 2\times224 \)
\( = 49 + 448 = 497 \)

Step2: Analyze the square pyramid

The square pyramid has a base side length \( s = 7 \) cm and slant height \( l = 6 \) cm (wait, no, the slant height? Wait, the height of the pyramid's triangular face? Wait, the pyramid's lateral faces are triangles with base 7 cm and height (slant height) let's check. Wait, the diagram shows 6 cm as the height of the pyramid? Wait, no, maybe the slant height? Wait, no, the problem: the pyramid is on top of the prism. The lateral surface area of the pyramid is \( 4\times\frac{1}{2}\times s\times l_{slant} \). Wait, maybe the 6 cm is the slant height? Wait, no, let's re - examine.
Wait, the rectangular prism: length = 7, width = 7, height = 16.
The pyramid: base is square with side 7, and the triangular faces: each has base 7 and height (the slant height) – wait, maybe the 6 cm is the height of the triangular face? Wait, no, maybe the height of the pyramid is 6? No, the surface area of the pyramid's lateral faces: for a square pyramid, lateral surface area \( LSA = 4\times\frac{1}{2}\times base\times slant\ height \).
Wait, maybe the 6 cm is the slant height. So each triangular face has area \( \frac{1}{2}\times7\times6 \), and there are 4 of them.
So lateral surface area of pyramid: \( 4\times\frac{1}{2}\times7\times6= 4\times21 = 84 \)

Step3: Calculate total surface area

Total surface area of the tower is the lateral surface area of the prism (excluding top) plus the lateral surface area of the pyramid.
So total \( = 497+84 = 581 \) square centimeters.

Answer:

581 square centimeters