Answer :
To solve this problem, let's understand the relationship given and how to use it to find the unknown.
The value of [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex] and inversely with the square of [tex]\( z \)[/tex]. This can be mathematically expressed as:
[tex]\[ y = k \cdot \frac{x}{z^2} \][/tex]
where [tex]\( k \)[/tex] is a constant.
### Step 1: Find the Constant [tex]\( k \)[/tex]
We use the first set of values provided to find [tex]\( k \)[/tex].
- When [tex]\( x = 24 \)[/tex], [tex]\( y = 6 \)[/tex], and [tex]\( z = 8 \)[/tex].
Plug these into our equation:
[tex]\[ 6 = k \cdot \frac{24}{8^2} \][/tex]
[tex]\[ 6 = k \cdot \frac{24}{64} \][/tex]
[tex]\[ 6 = k \cdot \frac{3}{8} \][/tex]
Solve for [tex]\( k \)[/tex]:
[tex]\[ k = 6 \cdot \frac{8}{3} \][/tex]
[tex]\[ k = 16 \][/tex]
### Step 2: Find the New Value of [tex]\( x \)[/tex]
Use the constant [tex]\( k \)[/tex] we found to solve for [tex]\( x \)[/tex] with the new conditions.
- Now, [tex]\( y = 20 \)[/tex] and [tex]\( z = 6 \)[/tex]. We need to find [tex]\( x \)[/tex].
[tex]\[ 20 = 16 \cdot \frac{x}{6^2} \][/tex]
[tex]\[ 20 = 16 \cdot \frac{x}{36} \][/tex]
To solve for [tex]\( x \)[/tex], multiply both sides by 36:
[tex]\[ 20 \cdot 36 = 16x \][/tex]
[tex]\[ 720 = 16x \][/tex]
Divide both sides by 16 to isolate [tex]\( x \)[/tex]:
[tex]\[ x = \frac{720}{16} \][/tex]
[tex]\[ x = 45 \][/tex]
Therefore, the value of [tex]\( x \)[/tex] when [tex]\( y = 20 \)[/tex] and [tex]\( z = 6 \)[/tex] is 45.
The value of [tex]\( y \)[/tex] varies directly with [tex]\( x \)[/tex] and inversely with the square of [tex]\( z \)[/tex]. This can be mathematically expressed as:
[tex]\[ y = k \cdot \frac{x}{z^2} \][/tex]
where [tex]\( k \)[/tex] is a constant.
### Step 1: Find the Constant [tex]\( k \)[/tex]
We use the first set of values provided to find [tex]\( k \)[/tex].
- When [tex]\( x = 24 \)[/tex], [tex]\( y = 6 \)[/tex], and [tex]\( z = 8 \)[/tex].
Plug these into our equation:
[tex]\[ 6 = k \cdot \frac{24}{8^2} \][/tex]
[tex]\[ 6 = k \cdot \frac{24}{64} \][/tex]
[tex]\[ 6 = k \cdot \frac{3}{8} \][/tex]
Solve for [tex]\( k \)[/tex]:
[tex]\[ k = 6 \cdot \frac{8}{3} \][/tex]
[tex]\[ k = 16 \][/tex]
### Step 2: Find the New Value of [tex]\( x \)[/tex]
Use the constant [tex]\( k \)[/tex] we found to solve for [tex]\( x \)[/tex] with the new conditions.
- Now, [tex]\( y = 20 \)[/tex] and [tex]\( z = 6 \)[/tex]. We need to find [tex]\( x \)[/tex].
[tex]\[ 20 = 16 \cdot \frac{x}{6^2} \][/tex]
[tex]\[ 20 = 16 \cdot \frac{x}{36} \][/tex]
To solve for [tex]\( x \)[/tex], multiply both sides by 36:
[tex]\[ 20 \cdot 36 = 16x \][/tex]
[tex]\[ 720 = 16x \][/tex]
Divide both sides by 16 to isolate [tex]\( x \)[/tex]:
[tex]\[ x = \frac{720}{16} \][/tex]
[tex]\[ x = 45 \][/tex]
Therefore, the value of [tex]\( x \)[/tex] when [tex]\( y = 20 \)[/tex] and [tex]\( z = 6 \)[/tex] is 45.