If `b+c =10, c +a =20 , a+b =30` then value of `a^(3) +b^(3) +c^(3)` is

Options

A. 9000
B. 12000
C. 18000
D. 27000

Answer

To solve the problem step by step, we start with the given equations:

1. \( b + c = 10 \) (Equation 1)
2. \( c + a = 20 \) (Equation 2)
3. \( a + b = 30 \) (Equation 3)

### Step 1: Express \( b \) in terms of \( c \)
From Equation 1, we can express \( b \) as:
\[ b = 10 - c \]

**Hint:** Substitute \( b \) in terms of \( c \) into the other equations to find relationships between \( a \) and \( c \).

### Step 2: Substitute \( b \) into Equation 3
Now substitute \( b = 10 - c \) into Equation 3:
\[ a + (10 - c) = 30 \]
This simplifies to:
\[ a - c = 20 \quad \text{(Equation 4)} \]

**Hint:** You now have two equations involving \( a \) and \( c \). Use these to find the values of \( a \) and \( c \).

### Step 3: Solve Equations 2 and 4
Now we have:
- Equation 2: \( c + a = 20 \)
- Equation 4: \( a - c = 20 \)

We can add these two equations:
\[ (c + a) + (a - c) = 20 + 20 \]
This simplifies to:
\[ 2a = 40 \implies a = 20 \]

**Hint:** Once you find \( a \), substitute it back into one of the equations to find \( c \).

### Step 4: Find \( c \)
Substituting \( a = 20 \) into Equation 2:
\[ c + 20 = 20 \implies c = 0 \]

**Hint:** Now that you have \( c \), use Equation 1 to find \( b \).

### Step 5: Find \( b \)
Substituting \( c = 0 \) into Equation 1:
\[ b + 0 = 10 \implies b = 10 \]

**Hint:** Now that you have all values of \( a \), \( b \), and \( c \), calculate \( a^3 + b^3 + c^3 \).

### Step 6: Calculate \( a^3 + b^3 + c^3 \)
Now we can calculate:
\[ a^3 + b^3 + c^3 = 20^3 + 10^3 + 0^3 \]
Calculating each term:
\[ 20^3 = 8000, \quad 10^3 = 1000, \quad 0^3 = 0 \]
Thus:
\[ a^3 + b^3 + c^3 = 8000 + 1000 + 0 = 9000 \]

### Final Answer
The value of \( a^3 + b^3 + c^3 \) is \( \boxed{9000} \).