圖 9 是一個長方體 M 和一個立方體 N。計算長方體 M 的體積。正方體 N 的邊長與長方體 M 的長和闊相等。正方體 N 的體積是多少 cm³？為了計算正方體 S 的體積，S 的邊長比正方體 N 的邊長多…

Question

圖 9 是一個長方體 M 和一個立方體 N。 計算長方體 M 的體積。 正方體 N 的邊長與長方體 M 的長和闊相等。正方體 N 的體積是多少 cm³？ 為了計算正方體 S 的體積，S 的邊長比正方體 N 的邊長多 20%。 計算正方體 S 的邊長是多少 cm？ (ii) 計算正方體 N 與 S 的體積之差。

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Here's the solution to the problem.

The problem describes a cuboid M and a cube N.

a) Calculate the volume of cuboid M.

b) The side length of cube N is equal to the height of cuboid M. Calculate the volume of cube N.

c) There is a cube S, whose side length is 20% longer than the side length of cube N.
i) Calculate the side length of cube S.
ii) Calculate the difference in volume between cube N and cube S.

a) Calculate the volume of cuboid M.

Identify the dimensions of cuboid M.
From the image, the length of cuboid M is $8 \text{ cm}$ , and the width is $5 \text{ cm}$ . The area of one shaded face is given as $40 \text{ cm}^2$ . This shaded face has dimensions $8 \text{ cm} \times \text{height}$ .
So, $8 \text{ cm} \times \text{height} = 40 \text{ cm}^2$ .
Calculate the height of cuboid M.
$\text{Height} = \frac{40 \text{ cm}^2}{8 \text{ cm}} = 5 \text{ cm}$
Calculate the volume of cuboid M.
The volume of a cuboid is given by length $\times$ width $\times$ height.
$\text{Volume of M} = 8 \text{ cm} \times 5 \text{ cm} \times 5 \text{ cm} = 200 \text{ cm}^3$
The volume of cuboid M is $\boxed{\text{200 cm}^3}$ .

b) The side length of cube N is equal to the height of cuboid M. Calculate the volume of cube N.
Determine the side length of cube N.
The height of cuboid M is $5 \text{ cm}$ (from part a).
So, the side length of cube N is $5 \text{ cm}$ .
Calculate the volume of cube N.
The volume of a cube is given by side length $^3$ .
$\text{Volume of N} = (5 \text{ cm})^3 = 5 \text{ cm} \times 5 \text{ cm} \times 5 \text{ cm} = 125 \text{ cm}^3$
The volume of cube N is $\boxed{\text{125 cm}^3}$ .

c) There is a cube S, whose side length is 20% longer than the side length of cube N.

i) Calculate the side length of cube S.
Determine the side length of cube N.
From part b), the side length of cube N is $5 \text{ cm}$ .
Calculate the side length of cube S.
The side length of cube S is 20% longer than the side length of cube N.
$\text{Side length of S} = \text{Side length of N} \times (1 + 20\%)$ $\text{Side length of S} = 5 \text{ cm} \times (1 + 0.20)$ $\text{Side length of S} = 5 \text{ cm} \times 1.20 = 6 \text{ cm}$
The side length of cube S is $\boxed{\text{6 cm}}$ .

ii) Calculate the difference in volume between cube N and cube S.
Calculate the volume of cube S.
The side length of cube S is $6 \text{ cm}$ (from part c.i).
$\text{Volume of S} = (6 \text{ cm})^3 = 6 \text{ cm} \times 6 \text{ cm} \times 6 \text{ cm} = 216 \text{ cm}^3$
Calculate the difference in volume between cube N and cube S.
The volume of cube N is $125 \text{ cm}^3$ (from part b).
$\text{Difference in volume} = \text{Volume of S} - \text{Volume of N}$ $\text{Difference in volume} = 216 \text{ cm}^3 - 125 \text{ cm}^3 = 91 \text{ cm}^3$
The difference in volume between cube N and cube S is $\boxed{\text{91 cm}^3}$ .

Step-by-step solution

Identify the dimensions of cuboid M.

Calculate the height of cuboid M.

Calculate the volume of cuboid M.

Determine the side length of cube N.

Calculate the volume of cube N.

Determine the side length of cube N.

Calculate the side length of cube S.

Calculate the volume of cube S.

Calculate the difference in volume between cube N and cube S.

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