Question

A three-level deck is being constructed at Hana's house. The highest level is in the shape of a rectangle whose length is twice its width. The middle level is a rectangle whose length is 4 feet longer than the length of the highest level, but whose width is 2 feet shorter than the width of the highest level. The lowest level of the deck is in the shape of a square, each side of which is as long as the width of the middle level. If x represents the width of the highest level, what is an algebraic expression that can represent the total surface area of the entire 3-deck structure? A. 5x²-4 B. 5x²-12 C. 5x²-4x-4 D. 5x²+4

Answer 1

The total surface area of the entire 3-deck structure is given by the expression 4x² + (2x + 4)(x - 2) + (x - 2)².

Step-by-step explanation:

The total surface area of the entire 3-deck structure can be calculated by finding the surface area of each level and adding them together. Let's break it down:

Level 1: Highest Level (Rectangle)

The length of the rectangle is twice its width. So, the length = 2x and the width = x. The surface area = 2(length × width) = 2(2x × x) = 4x².

Level 2: Middle Level (Rectangle)

The length is 4 feet longer than the length of the highest level, which is 2x + 4. The width is 2 feet shorter than the width of the highest level, which is x - 2. The surface area = (2x + 4)(x - 2).

Level 3: Lowest Level (Square)

The width of the square is the same as the width of the middle level, which is x - 2. The surface area = (x - 2)².

Total Surface Area:

To get the total surface area, add the surface areas of each level: 4x² + (2x + 4)(x - 2) + (x - 2)².