Certainly! The relationship described in your problem indicates that \( y \) is directly proportional to \( x^2 \). This can be expressed mathematically as:
\[ y = kx^2 \]
where \( k \) is the constant of proportionality. From the information given, we know that when \( x = 4 \), \( y = 64 \). We can use this information to find \( k \).
- Substitute \( x \) and \( y \) into the equation:
\[ 64 = k(4^2) \]
- Calculate \( 4^2 \):
\[ 64 = k(16) \]
- Solve for \( k \):
\[ k = \frac{64}{16} = 4 \]
Now that we have the value of \( k \), we can write the equation as:
\[ y = 4x^2 \]
Next, we need to find \( y \) when \( x = 6 \):
- Substitute \( x = 6 \) into the equation:
\[ y = 4(6^2) \]
- Calculate \( 6^2 \):
\[ y = 4(36) \]
- Finally, calculate \( y \):
\[ y = 144 \]
Thus, when \( x = 6 \), the value of \( y \) is 144.
In summary:
- \( k = 4 \)
- When \( x = 6 \), \( y = 144 \).