If `x+y=15`, then `(x-10)^(3)+(y-5)^(3)` is

Options

A. 25
B. 125
C. 625
D. 0

Answer

To solve the problem, we start with the given equation:

1. **Given**: \( x + y = 15 \)

We need to find the value of \( (x - 10)^3 + (y - 5)^3 \).

2. **Substituting**: We can express \( y \) in terms of \( x \):
\[ y = 15 - x \]

3. **Rewriting the expression**: Substitute \( y \) into the expression \( (x - 10)^3 + (y - 5)^3 \):
\[ (x - 10)^3 + ((15 - x) - 5)^3 \]
Simplifying the second term:
\[ (x - 10)^3 + (10 - x)^3 \]

4. **Using the identity**: We can use the identity for the sum of cubes:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Here, let \( a = x - 10 \) and \( b = 10 - x \). Notice that:
\[ a + b = (x - 10) + (10 - x) = 0 \]

5. **Conclusion**: Since \( a + b = 0 \), we can conclude:
\[ (x - 10)^3 + (10 - x)^3 = 0 \]

Thus, the value of \( (x - 10)^3 + (y - 5)^3 \) is \( 0 \).

### Final Answer:
\[ \boxed{0} \]