x varies directly as y and x varies inversely as the square of z. When y = 75 and x = 6, then z = 5. Find the value of x when y = 24 and z = 4 :

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem step by step, we will follow the relationships given in the question.

### Step 1: Understand the relationships
We know that:
- \( x \) varies directly as \( y \) (which means \( x \propto y \))
- \( x \) varies inversely as the square of \( z \) (which means \( x \propto \frac{1}{z^2} \))

Combining these two relationships, we can express \( x \) as:
\[ x = k \cdot \frac{y}{z^2} \]
where \( k \) is a constant.

### Step 2: Find the constant \( k \)
We have the values:
- \( y = 75 \)
- \( x = 6 \)
- \( z = 5 \)

Substituting these values into the equation:
\[ 6 = k \cdot \frac{75}{5^2} \]
Calculating \( 5^2 \):
\[ 5^2 = 25 \]
Now substituting back:
\[ 6 = k \cdot \frac{75}{25} \]
Calculating \( \frac{75}{25} \):
\[ \frac{75}{25} = 3 \]
So we have:
\[ 6 = k \cdot 3 \]
Now, solving for \( k \):
\[ k = \frac{6}{3} = 2 \]

### Step 3: Use the value of \( k \) to find \( x \) when \( y = 24 \) and \( z = 4 \)
Now we need to find \( x \) when:
- \( y = 24 \)
- \( z = 4 \)

Substituting these values into the equation:
\[ x = 2 \cdot \frac{24}{4^2} \]
Calculating \( 4^2 \):
\[ 4^2 = 16 \]
Now substituting back:
\[ x = 2 \cdot \frac{24}{16} \]
Calculating \( \frac{24}{16} \):
\[ \frac{24}{16} = \frac{3}{2} \]
Now substituting back:
\[ x = 2 \cdot \frac{3}{2} = 3 \]

### Final Answer
Thus, the value of \( x \) when \( y = 24 \) and \( z = 4 \) is:
\[ \boxed{3} \]