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2.
a) A cube and a cuboid have the same volume. The length of the cuboid is 3 cm bigger, the breadth is 2 cm bigger, and the height is 3 cm smaller than the edge of the cube.

b) Find the value of the edge of the cube using logarithms. Find the length of the cuboid.

Sagot :

GinnyAnsweridn
Sure, let's break down the problem and solve it step-by-step.

### Given:
1. The volume of the cube and the cuboid are the same.
2. The length of the cuboid is 3 cm more than the edge of the cube.
3. The breadth of the cuboid is 2 cm more than the edge of the cube.
4. The height of the cuboid is 3 cm less than the edge of the cube.

Let's denote the edge length of the cube by [tex]\( a \)[/tex].

### Step 1: Calculate the Volume of the Cube
The volume [tex]\( V_{cube} \)[/tex] of the cube is given by:
[tex]\[ V_{cube} = a^3 \][/tex]

### Step 2: Calculate the Dimensions and Volume of the Cuboid
- The length of the cuboid is [tex]\( a + 3 \)[/tex] cm.
- The breadth of the cuboid is [tex]\( a + 2 \)[/tex] cm.
- The height of the cuboid is [tex]\( a - 3 \)[/tex] cm.

The volume [tex]\( V_{cuboid} \)[/tex] of the cuboid is:
[tex]\[ V_{cuboid} = (a + 3)(a + 2)(a - 3) \][/tex]

### Step 3: Equating the Volumes
Since the cube and the cuboid have the same volume:
[tex]\[ a^3 = (a + 3)(a + 2)(a - 3) \][/tex]

### Step 4: Expanding the Right-Hand Side
To solve it, we expand the right-hand side:
[tex]\[ (a + 3)(a + 2)(a - 3) = (a + 3)(a^2 + 2a - 3a - 6) \][/tex]
[tex]\[ = (a + 3)(a^2 - a - 6) = a^3 - a^2 - 6a + 3a^2 - 3a - 18 = a^3 + 2a^2 - 9a - 18 \][/tex]

### Step 5: Forming the Equation
So, the equation becomes:
[tex]\[ a^3 = a^3 + 2a^2 - 9a - 18 \][/tex]

Subtract [tex]\( a^3 \)[/tex] from both sides:
[tex]\[ 0 = 2a^2 - 9a - 18 \][/tex]

We need to solve for [tex]\( a \)[/tex].

### Step 6: Solving the Quadratic Equation
Divide the equation by 2 to simplify (optional):
[tex]\[ 0 = a^2 - \frac{9}{2}a - 9 \][/tex]

### Step 7: Solving The Quadratic Equation Using the Python code:
By solving the quadratic equation, we find the real positive solution for [tex]\( a \)[/tex]:
[tex]\[ a = 6 \][/tex]

### Step 8: Finding the Dimensions of the Cuboid
- Length of the cuboid: [tex]\( a + 3 = 6 + 3 = 9 \)[/tex] cm
- Breadth of the cuboid: [tex]\( a + 2 = 6 + 2 = 8 \)[/tex] cm
- Height of the cuboid: [tex]\( a - 3 = 6 - 3 = 3 \)[/tex] cm

### Conclusion
The edge of the cube is 6 cm.
The dimensions of the cuboid are:
- Length: 9 cm
- Breadth: 8 cm
- Height: 3 cm
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