Question

quiz 3: volumes of solids
jayden and dakota are on the decorating committee for the prom. they need to cover the walls with fabric. if the room is 12 feet by 12 feet with a height of 9 feet, how much area will they cover if they cover the four walls?
432 ft²
576 ft²
648 ft²
720 ft²

Explanation:

Step1: Identify wall dimensions

The room is a square base (12 ft × 12 ft) with height 9 ft. There are 4 walls: 2 with dimensions \(12 \, \text{ft} \times 9 \, \text{ft}\) and 2 with \(12 \, \text{ft} \times 9 \, \text{ft}\) (since it's square, length and width are equal). Wait, actually, for a rectangular room (here square), the lateral surface area (walls) is calculated as \(2 \times (length \times height + width \times height)\). Since length = width = 12 ft, height = 9 ft.

Step2: Calculate area of one pair of walls

First pair (two walls): \(2 \times (12 \times 9) = 2 \times 108 = 216 \, \text{ft}^2\)

Step3: Calculate total area for four walls

Second pair (other two walls, same as first pair): also \(216 \, \text{ft}^2\)? Wait no, wait: \(2 \times (length \times height) + 2 \times (width \times height)\). Since length = width = 12, this is \(2 \times 12 \times 9 + 2 \times 12 \times 9 = 4 \times 12 \times 9\)? Wait no, no: \(2 \times (12 \times 9) + 2 \times (12 \times 9) = 4 \times 12 \times 9\)? Wait, no, actually, \(2 \times (length \times height + width \times height) = 2 \times height \times (length + width)\). Here, length = 12, width = 12, height = 9. So \(2 \times 9 \times (12 + 12) = 18 \times 24 = 432\)? Wait no, that's not right. Wait, no: length and width are both 12, so perimeter of the base is \(4 \times 12 = 48\) (since it's a square). Then lateral surface area is perimeter × height = \(48 \times 9 = 432 \, \text{ft}^2\)? Wait, but let's check again. Each wall: front and back are 12 (length) × 9 (height), left and right are 12 (width) × 9 (height). So front: 12×9, back: 12×9, left: 12×9, right: 12×9. So total: 4×(12×9) = 4×108 = 432. Wait, but the options have 432 as an option? Wait the options are 432, 576, 648, 720. Wait, wait, maybe I made a mistake. Wait, maybe the room is 12x12, height 9. Wait, no, maybe I misread the height? Wait the problem says "height of 9 feet"? Wait, let me re-express.

Wait, lateral surface area (walls) of a rectangular prism is \(2h(l + w)\). Here, l = 12, w = 12, h = 9. So \(2 \times 9 \times (12 + 12) = 18 \times 24 = 432\). So the area is 432 ft². Wait but let's check the options: first option is 432 ft². So that's the answer.

Answer:

432 ft² (the first option, 432 ft²)