The area of a side of a box is 120 sq. cm. The area of the other side of the box is 72 sq. cm. If the area of the upper surface of the box is 60 sq. cm. then find the volume of the box ?

Options

A. `259200" cm"^(3)`
B. `84000" cm"^(3)`
C. `86400" cm"^(3)`
D. `720" cm"^(3)`

Answer

To find the volume of the box given the areas of its sides, we can follow these steps:

### Step 1: Understand the given areas
We are given the areas of three sides of a cuboid:
- Area of one side (let's say Length × Height) = 120 sq. cm
- Area of another side (let's say Breadth × Height) = 72 sq. cm
- Area of the upper surface (Length × Breadth) = 60 sq. cm

### Step 2: Set up the equations
Let:
- Length = L
- Breadth = B
- Height = H

From the given areas, we can set up the following equations:
1. \( L \times H = 120 \) (1)
2. \( B \times H = 72 \) (2)
3. \( L \times B = 60 \) (3)

### Step 3: Express H in terms of L and B
From equation (1), we can express H:
\[ H = \frac{120}{L} \] (4)

From equation (2), we can express H as well:
\[ H = \frac{72}{B} \] (5)

### Step 4: Equate the two expressions for H
From equations (4) and (5):
\[ \frac{120}{L} = \frac{72}{B} \]

Cross-multiplying gives:
\[ 120B = 72L \]

### Step 5: Simplify the equation
Dividing both sides by 24:
\[ 5B = 3L \]
\[ B = \frac{3}{5}L \] (6)

### Step 6: Substitute B in equation (3)
Now substitute equation (6) into equation (3):
\[ L \times \left(\frac{3}{5}L\right) = 60 \]
\[ \frac{3}{5}L^2 = 60 \]

### Step 7: Solve for L
Multiplying both sides by 5:
\[ 3L^2 = 300 \]
Dividing by 3:
\[ L^2 = 100 \]
Taking the square root:
\[ L = 10 \text{ cm} \]

### Step 8: Find B using L
Using equation (6):
\[ B = \frac{3}{5} \times 10 = 6 \text{ cm} \]

### Step 9: Find H using L
Now substitute L back into equation (4) to find H:
\[ H = \frac{120}{10} = 12 \text{ cm} \]

### Step 10: Calculate the volume
The volume \( V \) of the cuboid is given by:
\[ V = L \times B \times H \]
Substituting the values:
\[ V = 10 \times 6 \times 12 \]
\[ V = 720 \text{ cm}^3 \]

### Final Answer
The volume of the box is **720 cm³**.

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