The volume of a cuboid is $120$ cubic meters, and its lateral surface area is $140$ square meters. The length, breadth, and height of the cuboid are $l$, $b$, and $h$ respectively, where all are integers and $l b$. Given the following quantities:

- Quantity I: $2l - b - \frac{h}{2}$
- Quantity II: $\frac{3b imes l}{h}$
- Quantity III: $2b - l + h$

Which of the following relations is true?

Question

The volume of a cuboid is $120$ cubic meters, and its lateral surface area is $140$ square meters. The length, breadth, and height of the cuboid are $l$, $b$, and $h$ respectively, where all are integers and $l > b$. Given the following quantities:

- Quantity I: $2l - b - \frac{h}{2}$
- Quantity II: $\frac{3b 	imes l}{h}$
- Quantity III: $2b - l + h$

Which of the following relations is true?

Accepted Answer

**Solution:**

1. Given the volume of the cuboid:
   $$l 	imes b 	imes h = 120$$

2. The lateral surface area is given by:
   $$2(l + b)h = 140 \implies (l + b)h = 70$$

3. Since $l$, $b$, and $h$ are integers and $l > b$, we find suitable values satisfying both equations.

4. Try $l = 4$, $b = 3$, then:
   - Volume check: $4 	imes 3 	imes h = 120 \implies h = 10$
   - Lateral surface check: $(4 + 3) 	imes 10 = 7 	imes 10 = 70$ (matches)

5. Calculate each quantity:
   - Quantity I:
     $$2l - b - \frac{h}{2} = 2(4) - 3 - \frac{10}{2} = 8 - 3 - 5 = 0$$

- Quantity II:
     $$\frac{3b 	imes l}{h} = \frac{3 	imes 3 	imes 4}{10} = \frac{36}{10} = 3.6$$

- Quantity III:
     $$2b - l + h = 2(3) - 4 + 10 = 6 - 4 + 10 = 12$$

6. Comparing the values:
   $$0 < 3.6 < 12$$

**Answer:** Quantity I $<$ Quantity II $<$ Quantity III

Answer

Answer

Answer