Use the given information to find the unknown value.

y  varies directly as the cube of  x . When  x=2 , then  y=5 . Find  y  when  x=3 .

To solve this problem, we can use the formula for direct variation. Since \( y \) varies directly as the cube of \( x \), we can express this relationship as:

\[
y = kx^3
\]

where \( k \) is the constant of proportionality.

### Step 1: Find the constant \( k \).
We know that when \( x = 2 \), \( y = 5 \). Substituting these values into the equation:

\[
5 = k(2^3)
\]

\[
5 = k(8)
\]

\[
k = \frac{5}{8}
\]

### Step 2: Find \( y \) when \( x = 3 \).
Now that we know \( k = \frac{5}{8} \), we can find \( y \) when \( x = 3 \) by substituting \( x = 3 \) into the equation:

\[
y = \frac{5}{8}(3^3)
\]

\[
y = \frac{5}{8}(27)
\]

\[
y = \frac{135}{8}
\]

\[
y = 16.875
\]

Thus, when \( x = 3 \), \( y \) is 16.875.

Would you like more details or have any questions about this problem?

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Here are some related questions for practice:

1. If \( y \) varies directly as the square of \( x \), and \( y = 12 \) when \( x = 4 \), find \( y \) when \( x = 5 \).
2. If \( y \) varies directly as \( x \) and \( y = 20 \) when \( x = 4 \), find \( y \) when \( x = 10 \).
3. If \( y \) varies directly as the fourth power of \( x \) and \( y = 81 \) when \( x = 3 \), find \( y \) when \( x = 2 \).
4. Suppose \( y \) varies inversely as \( x \), and \( y = 10 \) when \( x = 2 \). Find \( y \) when \( x = 5 \).
5. If \( y \) varies directly as \( x^2 \) and \( y = 25 \) when \( x = 5 \), find \( x \) when \( y = 100 \).
6. If \( y \) varies directly as the square root of \( x \), and \( y = 7 \) when \( x = 49 \), find \( y \) when \( x = 16 \).
7. If \( y \) varies directly as \( x^3 \) and \( y = 64 \) when \( x = 4 \), find \( y \) when \( x = 2 \).
8. Suppose \( y \) varies directly as \( x^n \) and \( y = 32 \) when \( x = 2 \) for a certain \( n \). Find the value of \( n \) if \( y = 512 \) when \( x = 4 \).

**Tip:** In direct variation problems, finding the constant of proportionality first can simplify solving the problem.

Learn how to solve problems involving direct variation where y varies directly as the cube of x. Practice finding the constant of proportionality and calculating y for different values of x. Detailed step-by-step solution provided.

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