The cost of painting the surface of a solid cuboid of dimensions 40 cm × 36 cm x x cm at ₹ 20 per 100 cm$^2$ is ₹ 1,032. What is the value of x ?

Q: The cost of painting the surface of a solid cuboid of dimensions 40 cm × 36 cm x x cm at ₹ 20 per 100 cm$^2$ is ₹ 1,032. What is the value of x ?

To determine the value of $ x $ for the given cuboid, we need to calculate the total surface area of the cuboid and relate it to the cost of painting. ### Explanation: 1. **Surface Area Calculation:** The surface area $ S $ of a cuboid with dimensions $ l $, $ b $, and $ h $ is given by: \[ S = 2(lb + bh + hl) \] For the given cuboid, $ l = 40 $ cm, $ b = 36 $ cm, and $ h = x $ cm. Therefore, the surface area is: \[ S = 2(40 \times 36 + 36 \times x + x \times 40) \] Simplifying, we get: \[ S = 2(1440 + 36x + 40x) = 2(1440 + 76x) \] \[ S = 2880 + 152x \] 2. **Cost Calculation:** The cost of painting is given as ₹ 20 per 100 cm$^2$. The total cost is ₹ 1,032. Therefore, the equation for the cost is: \[ \frac{S}{100} \times 20 = 1032 \] Substituting the expression for $ S $: \[ \frac{2880 + 152x}{100} \times 20 = 1032 \] Simplifying: \[ (2880 + 152x) \times 0.2 = 1032 \] \[ 2880 + 152x = 5160 \] \[ 152x = 5160 - 2880 \] \[ 152x = 2280 \] \[ x = \frac{2280}{152} \] \[ x = 15 \] 3. **Conclusion:** The calculated value of $ x $ is 15, which corresponds to Option 2. Therefore, Option 2 is the correct answer. ### Explanation for Incorrect Options: - **Option 1 (20):** If $ x = 20 $, the surface area would be larger, leading to a higher cost than ₹ 1,032, which contradicts the given cost. - **Option 3 (24):** Similarly, if $ x = 24 $, the surface area and thus the cost would exceed ₹ 1,032. - **Option 4 (12):** If $ x = 12 $, the surface area would be smaller, resulting in a cost less than ₹ 1,032. Thus, the only value of $ x $ that satisfies the given conditions is 15, making Option 2 the correct choice.

Question

Accepted Answer

To determine the value of $ x $ for the given cuboid, we need to calculate the total surface area of the cuboid and relate it to the cost of painting.

### Explanation:

1. **Surface Area Calculation:**
   The surface area $ S $ of a cuboid with dimensions $ l $, $ b $, and $ h $ is given by:
   $$
   S = 2(lb + bh + hl)
   $$
   For the given cuboid, $ l = 40 $ cm, $ b = 36 $ cm, and $ h = x $ cm. Therefore, the surface area is:
   $$
   S = 2(40 \times 36 + 36 \times x + x \times 40)
   $$
   Simplifying, we get:
   $$
   S = 2(1440 + 36x + 40x) = 2(1440 + 76x)
   $$
   $$
   S = 2880 + 152x
   $$

2. **Cost Calculation:**
   The cost of painting is given as ₹ 20 per 100 cm$^2$. The total cost is ₹ 1,032. Therefore, the equation for the cost is:
   $$
   \frac{S}{100} \times 20 = 1032
   $$
   Substituting the expression for $ S $:
   $$
   \frac{2880 + 152x}{100} \times 20 = 1032
   $$
   Simplifying:
   $$
   (2880 + 152x) \times 0.2 = 1032
   $$
   $$
   2880 + 152x = 5160
   $$
   $$
   152x = 5160 - 2880
   $$
   $$
   152x = 2280
   $$
   $$
   x = \frac{2280}{152}
   $$
   $$
   x = 15
   $$

3. **Conclusion:**
   The calculated value of $ x $ is 15, which corresponds to Option 2. Therefore, Option 2 is the correct answer.

### Explanation for Incorrect Options:

- **Option 1 (20):** If $ x = 20 $, the surface area would be larger, leading to a higher cost than ₹ 1,032, which contradicts the given cost.
- **Option 3 (24):** Similarly, if $ x = 24 $, the surface area and thus the cost would exceed ₹ 1,032.
- **Option 4 (12):** If $ x = 12 $, the surface area would be smaller, resulting in a cost less than ₹ 1,032.

Thus, the only value of $ x $ that satisfies the given conditions is 15, making Option 2 the correct choice.