Suppose p, q, r are such that `p+q=r and pqr = 30`, then what is the value of `p^(3)+q^(3)-r^(3)`?

Options

A. 0
B. 90
C. `-90`
D. cannot be determined with given data

Answer

To solve the problem, we need to find the value of \( p^3 + q^3 - r^3 \) given that \( p + q = r \) and \( pqr = 30 \).

### Step-by-Step Solution:

1. **Use the identity for the sum of cubes**:
We know that:
\[ p^3 + q^3 = (p + q)(p^2 - pq + q^2) \]
Since \( p + q = r \), we can rewrite this as:
\[ p^3 + q^3 = r(p^2 - pq + q^2) \]

2. **Express \( p^2 - pq + q^2 \)**:
We can express \( p^2 - pq + q^2 \) in terms of \( p + q \) and \( pq \):
\[ p^2 + q^2 = (p + q)^2 - 2pq = r^2 - 2pq \]
Thus,
\[ p^2 - pq + q^2 = (r^2 - 2pq) - pq = r^2 - 3pq \]

3. **Substituting back into the equation**:
Now substituting back into the equation for \( p^3 + q^3 \):
\[ p^3 + q^3 = r(r^2 - 3pq) \]

4. **Now, we need to find \( p^3 + q^3 - r^3 \)**:
We can express \( r^3 \) as:
\[ r^3 = r \cdot r^2 \]
Therefore,
\[ p^3 + q^3 - r^3 = r(r^2 - 3pq) - r^3 \]
Simplifying this gives:
\[ p^3 + q^3 - r^3 = r(r^2 - 3pq - r^2) = -3r \cdot pq \]

5. **Substituting \( pq \)**:
From the given \( pqr = 30 \):
\[ pq = \frac{30}{r} \]
Substituting this into the equation:
\[ p^3 + q^3 - r^3 = -3r \cdot \frac{30}{r} = -90 \]

### Final Answer:
Thus, the value of \( p^3 + q^3 - r^3 \) is \(-90\).