The figure shows an open box in the shape of a cuboid. The box is made of wooden boards of 2 cm thick. (a) Find the capacity of the box. (b) Find the volume of wooden boards used.
Understand the Problem
The question describes an open wooden box in the shape of a cuboid with a specified thickness for the wooden boards. Part (a) requires us to find the internal volume (capacity) of the box. To compute this, we will need to subtract the thickness of the wood from the overall dimensions. Part (b) asks for the volume of the wood used to make the box. To compute this, we will subtract the internal volume from the external volume.
Answer
(a) $9828 \text{ cm}^3$ (b) $5172 \text{ cm}^3$
Answer for screen readers
(a) The capacity of the box is $9828 \text{ cm}^3$.
(b) The volume of wooden boards used is $5172 \text{ cm}^3$.
Steps to Solve
- Calculate the internal length
The external length is 30 cm and the wood thickness is 2 cm on each side. The internal length is therefore:
$30 - 2 - 2 = 26$ cm
- Calculate the internal width
The external width is 25 cm, and the wood thickness is 2 cm on each side. The internal width is therefore:
$25 - 2 - 2 = 21$ cm
- Calculate the internal height
The external height is 20 cm. Since the box is open at the top, the wood thickness only matters at the bottom. The internal height is therefore:
$20 - 2 = 18$ cm
- Calculate the internal volume (capacity)
The capacity of the box is the internal volume, which is length $\times$ width $\times$ height:
$26 \times 21 \times 18 = 9828 \text{ cm}^3$
- Calculate the external volume
The external volume is length $\times$ width $\times$ height:
$30 \times 25 \times 20 = 15000 \text{ cm}^3$
- Calculate the volume of the wood
The volume of the wood is the external volume minus the internal volume:
$15000 - 9828 = 5172 \text{ cm}^3$
(a) The capacity of the box is $9828 \text{ cm}^3$.
(b) The volume of wooden boards used is $5172 \text{ cm}^3$.
More Information
The internal volume represents the space inside the box that can be filled. The volume of the wood represents the amount of material used to construct the box.
Tips
A common mistake is forgetting to subtract the thickness of the wood from both sides of the length and width when calculating the internal dimensions. Also, students may subtract the thickness from both the top and bottom when calculating the internal height, but since the box is open, the thickness should only be subtracted from the bottom.
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