Question

Liu has 144 one -centimetre cubes. He arranges all of the cubes into a cuboid. The perimeter of the top of the cuboid is 22cm. Each side of the cuboid has a length greater than 2 Find the height of the cuboid.

Answer 1

To find the height of the cuboid, use the formula for volume and solve for the height using the given information.

Step-by-step explanation:

To find the height of the cuboid, we can use the formula for the volume of a cuboid: V = lwh, where l is the length, w is the width, and h is the height.

We are given the perimeter of the top face of the cuboid, which is 22 cm. Since each side of the cuboid has a length greater than 2, we can use this information to find the length and width of the top face.

Let the length of the top face be x cm. Then the width of the top face would be (22 - 4x) cm. Since the cuboid is made up of 144 cubes, the volume can be expressed as V = x(22 - 4x)h = 144.

We can solve this equation for x and then substitute it back into the equation to find the value of h.

Answer 2

By considering the perimeter of 22cm and the total volume of 144 cm³, we can determine that the dimensions that satisfy these conditions with a length greater than 2 cm are 7cm by 4cm, leading to a height of 5 cm for the cuboid formed by the one-centimetre cubes.

Step-by-step explanation:

Liu has 144 one-centimetre cubes and arranges them into a cuboid with the perimeter of the top being 22cm. We know that the perimeter (P) of a rectangle (which is the shape of the top of the cuboid) is given by P = 2(length + width). Since each side must be greater than 2 cm, let's consider the possible dimensions (length and width) that give us a perimeter of 22cm. One possible dimension combination is 7cm (length) and 4cm (width), since 2(7cm + 4cm) equals 22cm.

Knowing the total number of cubes (volume) is 144 cm³ and having determined the possible dimensions of the base, we can now find the height. The volume (V) of a cuboid is calculated by:

V = length x width x height.

Therefore, the height can be found by dividing the volume by the product of length and width. If the length is 7cm and width is 4cm, the volume is 144 cm³.

The height (h) would be:

h = V / (length x width) = 144 cm³ / (7cm x 4cm) = 144 cm³ / 28 cm² = 5.14 cm.

Since all measurements must be whole numbers (because the cubes are 1 cm³), the closest whole number for height that maintains a volume of 144 cm³ is 5 cm.

Therefore, the height of the cuboid is 5 cm.