Question

I'm studying algebra and I came across a problem that involves direct proportions. The problem states that a quantity 'y' is directly proportional to 'x' squared. If y equals 64 when x equals 4, I need to find the value of y when x equals 6. Can you find a similar problem on this Q&A forum and provide the solution?

Answer 1

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 \( k \).

1. Substitute \( x \) and \( y \) into the equation:

\[
64 = k(4^2)
\]

2. Calculate \( 4^2 \):

\[
64 = k(16)
\]

3. 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 \( x = 6 \):

1. Substitute \( x = 6 \) into the equation:

\[
y = 4(6^2)
\]

2. Calculate \( 6^2 \):

\[
y = 4(36)
\]

3. Finally, calculate \( y \):

\[
y = 144
\]

Thus, when \( x = 6 \), the value of \( y \) is **144**.

In summary:
- \( k = 4 \)
- When \( x = 6 \), \( y = 144 \).