A box made of wood of uniform thickness, 2cm has external dimensions 25cm x 20cm x 12cm. Find the volume of wood and the cost of painting all its surfaces at the rate Rs. 1.5 per cm^2 ? A. 3123 cm^3 and Rs. 6015 B. 3312 cm^3 and Rs. 5016 C. 2133 cm^3 and Rs. 5160 D. None of above

We can break this problem into two parts:

1. **Finding the volume of the wood**:  
The box has uniform thickness, meaning it has both external and internal dimensions. We will calculate the volumes of the external and internal boxes and subtract them to get the volume of the wood.

2. **Calculating the cost of painting the surfaces**:  
We will calculate the surface area of the external box and multiply it by the rate per cm² to find the cost of painting.

### Step 1: Volume of Wood
- External dimensions of the box:  
  \( 25 \, \text{cm} \times 20 \, \text{cm} \times 12 \, \text{cm} \)

  The **volume of the external box** is:  
  \[
  V_{\text{external}} = 25 \times 20 \times 12 = 6000 \, \text{cm}^3
  \]

- Internal dimensions of the box:  
  Since the wood has a thickness of 2 cm, we subtract 2 cm from each side (both ends) of the external dimensions to get the internal dimensions:  
  Internal length = \( 25 - 2 \times 2 = 21 \, \text{cm} \)  
  Internal width = \( 20 - 2 \times 2 = 16 \, \text{cm} \)  
  Internal height = \( 12 - 2 \times 2 = 8 \, \text{cm} \)

  The **volume of the internal box** is:  
  \[
  V_{\text{internal}} = 21 \times 16 \times 8 = 2688 \, \text{cm}^3
  \]

- The **volume of the wood** is:  
  \[
  V_{\text{wood}} = V_{\text{external}} - V_{\text{internal}} = 6000 - 2688 = 3312 \, \text{cm}^3
  \]

### Step 2: Cost of Painting
- The **surface area** of the external box (including all sides) is given by the formula for the surface area of a rectangular box:
  \[
  A = 2 \left( lw + lh + wh \right)
  \]
  where \( l = 25 \, \text{cm}, w = 20 \, \text{cm}, h = 12 \, \text{cm} \).

  Substituting the values:
  \[
  A = 2 \left( 25 \times 20 + 25 \times 12 + 20 \times 12 \right)
  = 2 \left( 500 + 300 + 240 \right)
  = 2 \times 1040 = 2080 \, \text{cm}^2
  \]

- The cost of painting is:
  \[
  \text{Cost} = 2080 \times 1.5 = 3120 \, \text{Rs.}
  \]

### Final Answer:
- The volume of wood is \( 3312 \, \text{cm}^3 \).
- The cost of painting is Rs. 3120.

Thus, the correct answer is **none of the above**, as the cost doesn't match any given options.

---

Would you like further details on any part of this solution? Here are 5 related questions you might explore:

1. How would the solution change if the wood thickness were increased to 3 cm?
2. What is the total internal surface area of the box, excluding the top?
3. How can the cost of painting the box vary if the rate changes to Rs. 2 per cm²?
4. What is the impact of different material thicknesses on the volume of wood in cubic meters?
5. How could you calculate the cost for painting only the external faces of the box?

**Tip**: When working with surface area, always confirm whether the problem asks for painting internal, external, or both surfaces!

This problem involves finding the volume of wood in a wooden box with external dimensions of 25cm x 20cm x 12cm and a thickness of 2cm, as well as calculating the cost of painting its surfaces at Rs. 1.5 per cm². Step-by-step solution included for students in Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation