If ` 16 x^(2) + y^(2) = 48 and xy = 11 ` find the value of ` 64 x^(3) + y^(3)`

Options

A. ` 4 sqrt(34)`
B. ` 8 sqrt(36)`
C. ` 4 sqrt(36`
D. ` 8 sqrt(34)`

Answer

To solve the problem, we need to find the value of \( 64x^3 + y^3 \) given the equations \( 16x^2 + y^2 = 48 \) and \( xy = 11 \).

### Step-by-Step Solution:

1. **Rewrite the first equation**:
We can express \( 16x^2 + y^2 = 48 \) in a different form:
\[ 16x^2 + y^2 = 48 \implies 4^2 x^2 + y^2 = 48 \]
This can be rewritten as:
\[ (4x)^2 + y^2 = 48 \]

2. **Use the identity for squares**:
We know that:
\[ (a + b)^2 = a^2 + b^2 + 2ab \]
Let \( a = 4x \) and \( b = y \). Then we can express:
\[ (4x + y)^2 = (4x)^2 + y^2 + 2(4x)(y) \]
Substituting the known values:
\[ (4x + y)^2 = 48 + 2(4xy) \]

3. **Substitute \( xy \)**:
Since \( xy = 11 \), we can substitute this into the equation:
\[ (4x + y)^2 = 48 + 2(4 \cdot 11) = 48 + 88 = 136 \]

4. **Take the square root**:
Taking the square root of both sides gives:
\[ 4x + y = \sqrt{136} = 2\sqrt{34} \]

5. **Use the formula for cubes**:
We need to find \( 64x^3 + y^3 \). We can use the identity:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Here, let \( a = 4x \) and \( b = y \). Thus:
\[ 64x^3 + y^3 = (4x)^3 + y^3 = (4x + y)((4x)^2 - (4x)(y) + y^2) \]

6. **Substitute known values**:
We already found \( 4x + y = 2\sqrt{34} \). Now we need to calculate \( (4x)^2 - (4x)(y) + y^2 \):
- \( (4x)^2 = 16x^2 \)
- \( (4x)(y) = 4xy = 4 \cdot 11 = 44 \)
- \( y^2 = y^2 \)

Therefore:
\[ (4x)^2 - (4x)(y) + y^2 = 16x^2 - 44 + y^2 \]

7. **Substitute \( 16x^2 + y^2 \)**:
From the first equation, we know \( 16x^2 + y^2 = 48 \):
\[ 16x^2 - 44 + y^2 = 48 - 44 = 4 \]

8. **Final calculation**:
Now substituting back:
\[ 64x^3 + y^3 = (4x + y)(4) = (2\sqrt{34})(4) = 8\sqrt{34} \]

Thus, the value of \( 64x^3 + y^3 \) is \( 8\sqrt{34} \).