What are the dimensions (length, breadth and height, respectively) of a cuboid with volume 720 cu cm, surface area 484 sq cm and the area of the base 72 sq cm?

Options

A. a. 9,8 and 10 cm
B. c. 12, 6 and 10 cm
C. b. and 10 cm
D. d. 30, 2 and 12 cm

Answer

To find the dimensions (length, breadth, and height) of a cuboid given its volume, surface area, and area of the base, we can follow these steps:

### Step 1: Identify the given values
- Volume (V) = 720 cm³
- Surface Area (SA) = 484 cm²
- Area of the base (A) = 72 cm²

### Step 2: Use the area of the base to find the height
The area of the base of a cuboid is given by:
\[ A = \text{length} \times \text{breadth} \]
Let the length be \( l \) and the breadth be \( b \). Therefore:
\[ l \times b = 72 \quad \text{(1)} \]

From the volume formula:
\[ V = l \times b \times h \]
We can express the height \( h \) as:
\[ h = \frac{V}{A} = \frac{720}{72} = 10 \, \text{cm} \quad \text{(2)} \]

### Step 3: Substitute the height into the surface area formula
The surface area of a cuboid is given by:
\[ SA = 2(lb + bh + hl) \]
Substituting \( h = 10 \) into the surface area formula:
\[ 484 = 2(lb + 10b + 10l) \]
Dividing both sides by 2:
\[ 242 = lb + 10b + 10l \quad \text{(3)} \]

### Step 4: Substitute \( lb \) from equation (1) into equation (3)
From equation (1), we know:
\[ lb = 72 \]
Substituting this into equation (3):
\[ 242 = 72 + 10b + 10l \]
Rearranging gives:
\[ 10b + 10l = 242 - 72 \]
\[ 10b + 10l = 170 \]
Dividing by 10:
\[ b + l = 17 \quad \text{(4)} \]

### Step 5: Solve the system of equations
Now we have two equations:
1. \( l \times b = 72 \) (equation 1)
2. \( l + b = 17 \) (equation 4)

Let’s express \( l \) in terms of \( b \) from equation (4):
\[ l = 17 - b \]
Now substitute this into equation (1):
\[ (17 - b) \times b = 72 \]
Expanding this:
\[ 17b - b^2 = 72 \]
Rearranging gives:
\[ b^2 - 17b + 72 = 0 \]

### Step 6: Solve the quadratic equation
To solve \( b^2 - 17b + 72 = 0 \), we can factor it:
\[ (b - 8)(b - 9) = 0 \]
Thus, \( b = 8 \) or \( b = 9 \).

### Step 7: Find the corresponding length
If \( b = 8 \):
\[ l = 17 - 8 = 9 \]
If \( b = 9 \):
\[ l = 17 - 9 = 8 \]
Thus, the dimensions are:
- Length \( l = 9 \) cm
- Breadth \( b = 8 \) cm
- Height \( h = 10 \) cm

### Final Answer:
The dimensions of the cuboid are:
- Length = 9 cm
- Breadth = 8 cm
- Height = 10 cm

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