If the sum of the length, breadth and depth of a cuboid is 20 cm and its diagonal is 4`sqrt5 `cm, then its surface area is

Options

A. 400 `cm^2`
B. 420 `cm^2`
C. 300 `cm^2`
D. 320 `cm^2`

Answer

To solve the problem step by step, we will use the given information about the cuboid's dimensions and its diagonal.

### Step 1: Set up the equations
We are given that the sum of the length (L), breadth (B), and height (H) of the cuboid is 20 cm:
\[ L + B + H = 20 \quad \text{(Equation 1)} \]

We are also given that the diagonal (D) of the cuboid is \( 4\sqrt{5} \) cm. The formula for the diagonal of a cuboid is:
\[ D = \sqrt{L^2 + B^2 + H^2} \]
Thus, we have:
\[ \sqrt{L^2 + B^2 + H^2} = 4\sqrt{5} \]
Squaring both sides gives us:
\[ L^2 + B^2 + H^2 = (4\sqrt{5})^2 = 80 \quad \text{(Equation 2)} \]

### Step 2: Square Equation 1
Now, we will square Equation 1:
\[ (L + B + H)^2 = 20^2 \]
Using the identity \( (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + ac + bc) \), we get:
\[ L^2 + B^2 + H^2 + 2(LB + BH + HL) = 400 \]

### Step 3: Substitute Equation 2 into the squared equation
From Equation 2, we know that \( L^2 + B^2 + H^2 = 80 \). Substituting this into the squared equation gives:
\[ 80 + 2(LB + BH + HL) = 400 \]

### Step 4: Solve for LB + BH + HL
Now, we can isolate \( LB + BH + HL \):
\[ 2(LB + BH + HL) = 400 - 80 \]
\[ 2(LB + BH + HL) = 320 \]
Dividing both sides by 2:
\[ LB + BH + HL = 160 \quad \text{(Equation 3)} \]

### Step 5: Calculate the surface area
The surface area (SA) of a cuboid is given by the formula:
\[ SA = 2(LB + BH + HL) \]
Substituting Equation 3 into this formula:
\[ SA = 2 \times 160 = 320 \, \text{cm}^2 \]

### Final Answer
Thus, the surface area of the cuboid is:
\[ \boxed{320 \, \text{cm}^2} \]