the walls, floor, and ceiling of a room form a rectangular solid. The total surface are of the room is 992 square feet. The dimensions of the floor are 12 feet by 20 feet

a. What is the lateral area of the room
b. What is the height of the room.

As = 2*(W*h)) + 2*(L*h) + 2(L*W)

a. As = Al + 2*(L*W) = 992 Ft^2
Al + 2*(20*12) = 992
Al = 992 - 480 = 512 Ft^2.=Lateral area.

b. Al = 2(W*h) + 2(L*h) = 512 Ft^2
2(12*h) + 2(20*h) = 512
Solve for h.

To solve this problem, we need to understand the concept of lateral area and how it relates to the total surface area of a rectangular solid.

a. Lateral area is the total surface area of the room excluding the floor and ceiling. It is calculated by summing up the areas of the four walls. Since the room is a rectangular solid, we can observe that the lateral area is equal to the total surface area minus the area of the floor and ceiling.

b. The height of the room is the length of one of its walls.

Now, let's find the answers to the given questions.

a. To calculate the lateral area, we need to subtract the area of the floor and ceiling from the total surface area. Given that the total surface area is 992 square feet and the dimensions of the floor are 12 feet by 20 feet, we can calculate the area of the floor using the formula: area = length × width.
The area of the floor is 12 feet × 20 feet = 240 square feet.
The area of the ceiling is also the same as the floor, so the combined area of the floor and ceiling is 2 times the area of the floor, which is 2 × 240 square feet = 480 square feet.
Therefore, the lateral area is 992 square feet - 480 square feet = 512 square feet.

b. Since the room is a rectangular solid and the dimensions of the floor are given as 12 feet by 20 feet, the height of the room is the length of one of its walls.
Therefore, the height of the room is 12 feet.

To summarize:
a. The lateral area of the room is 512 square feet.
b. The height of the room is 12 feet.