The height of a rectangular solid is 5 times its width and its length is 8 times its height . If the volume of the walls is ` 102 . 4 cm ^(3)` , find its length

Answer

To solve the problem step by step, we will follow the given information and use algebra to find the length of the rectangular solid.

### Step-by-Step Solution:

1. **Define Variables**:
Let the width of the rectangular solid be \( x \) cm.

2. **Express Height and Length**:
- The height \( h \) is given as 5 times the width:
\[ h = 5x \]
- The length \( l \) is given as 8 times the height:
\[ l = 8h = 8(5x) = 40x \]

3. **Volume of the Rectangular Solid**:
The volume \( V \) of a rectangular solid is calculated using the formula:
\[ V = l \times w \times h \]
Substituting the expressions for length, width, and height:
\[ V = (40x) \times x \times (5x) = 200x^3 \]

4. **Set Up the Equation**:
We know the volume of the walls is given as \( 102.4 \, \text{cm}^3 \). Therefore, we set up the equation:
\[ 200x^3 = 102.4 \]

5. **Solve for \( x^3 \)**:
To isolate \( x^3 \), divide both sides by 200:
\[ x^3 = \frac{102.4}{200} \]
Simplifying the right side:
\[ x^3 = \frac{1024}{2000} = \frac{512}{1000} \]

6. **Finding the Cube Root**:
To find \( x \), we take the cube root of both sides:
\[ x = \sqrt[3]{\frac{512}{1000}} = \frac{\sqrt[3]{512}}{\sqrt[3]{1000}} = \frac{8}{10} = 0.8 \, \text{cm} \]

7. **Calculate the Length**:
Now that we have the width, we can find the length:
\[ l = 40x = 40 \times 0.8 = 32 \, \text{cm} \]

### Final Answer:
The length of the rectangular solid is \( 32 \, \text{cm} \).

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