Answer: Let's tackle each question step by step:
Find the length of a side of the cube:
In a cube, all sides are equal in length. So, to find the length of a side, we can use any given measurement.
Let's assume the length of a side of the cube is "s" meters.
The formula for the volume of a cube is given by:
Volume = side^3
Since all sides of the cube are equal, we can express the volume as:
Volume = s^3
Now, let's find the volume of the cube using the given information:
Given: Volume = 216 m³
s^3 = 216
To find the length of a side (s), we take the cube root of both sides:
s = ∛(216)
s ≈ 6 meters
So, the length of a side of the cube is approximately 6 meters.
Find the height of a room:
Let's assume the height of the room is "h" meters.
The given information tells us that the perimeter of the base (the bottom of the room) is 60 meters. Since the base of the room is rectangular, the perimeter is the sum of all four sides of the rectangle.
Perimeter of the base = 60 meters
The formula for the perimeter of a rectangle is given by:
Perimeter = 2 × (length + width)
In our case, since the room is vertical, the length and width refer to the sides of the base.
Given: Perimeter = 60 meters
We know that the area of the four walls is 240 cm². Since there are four walls, the total area of the four walls is the sum of the areas of each wall.
Total area of four walls = 240 cm²
Let's convert this area to square meters since the perimeter is in meters.
1 cm² = 0.0001 m²
240 cm² = 240 × 0.0001 m² = 0.024 m²
The formula for the total area of the four walls of a rectangular room is given by:
Total area of four walls = 2 × (height × width + height × length)
Given: Total area of four walls = 0.024 m²
Since the height of the room (h) is the unknown we want to find, we need to solve for it:
0.024 = 2 × (h × width + h × length)
Substitute the given perimeter (60 meters) into the formula for the perimeter of a rectangle:
60 = 2 × (h + width + h + length)
Since we know the length and width of the base are equal in a rectangular room, we can simplify further:
60 = 4 × (h + length)
Now, solve for the height (h):
h + length = 60 / 4
h + length = 15
h = 15 - length
Substitute the length of the base (width) into the formula:
h = 15 - width
Since the height and width of the base are equal, we get:
h = 15 - width = 15 - width
Now, we know that the width of the base is equal to the length of the base in a square room. Since we found the length of the side of the cube to be 6 meters earlier, the width (and height) of the base of the room is also 6 meters.
h = 15 - 6 = 9 meters
So, the height of the room is 9 meters.