If `yx^(2)/(3)`="constant"
when` x=x_(1)`, then y=300, then find the y when `x=(8x_(1))/(27)`

Options

A. 450
B. 375
C. 675
D. 405

Answer

To solve the problem step by step, we will follow the given information and derive the required value of \( y \) when \( x = \frac{8x_1}{27} \).

### Step 1: Understand the relationship
We are given that:
\[ y \cdot x^{\frac{2}{3}} = \text{constant} \]
Let’s denote this constant as \( k \).

### Step 2: Find the constant \( k \)
When \( x = x_1 \), \( y = 300 \):
\[ 300 \cdot (x_1)^{\frac{2}{3}} = k \]
Thus, we can express \( k \) as:
\[ k = 300 \cdot (x_1)^{\frac{2}{3}} \]

### Step 3: Substitute the new value of \( x \)
Now we need to find \( y \) when \( x = \frac{8x_1}{27} \). We substitute this value into the equation:
\[ y \cdot \left(\frac{8x_1}{27}\right)^{\frac{2}{3}} = k \]

### Step 4: Calculate \( \left(\frac{8x_1}{27}\right)^{\frac{2}{3}} \)
First, we simplify \( \left(\frac{8}{27}\right)^{\frac{2}{3}} \):
\[ \left(\frac{8}{27}\right)^{\frac{2}{3}} = \frac{8^{\frac{2}{3}}}{27^{\frac{2}{3}}} = \frac{(2^3)^{\frac{2}{3}}}{(3^3)^{\frac{2}{3}}} = \frac{2^2}{3^2} = \frac{4}{9} \]
Thus,
\[ \left(\frac{8x_1}{27}\right)^{\frac{2}{3}} = \frac{4}{9} (x_1)^{\frac{2}{3}} \]

### Step 5: Substitute back into the equation
Now substituting back into the equation:
\[ y \cdot \frac{4}{9} (x_1)^{\frac{2}{3}} = 300 \cdot (x_1)^{\frac{2}{3}} \]

### Step 6: Solve for \( y \)
We can cancel \( (x_1)^{\frac{2}{3}} \) from both sides (assuming \( x_1 \neq 0 \)):
\[ y \cdot \frac{4}{9} = 300 \]
Now, solve for \( y \):
\[ y = 300 \cdot \frac{9}{4} = 300 \cdot 2.25 = 675 \]

### Final Answer
Thus, the value of \( y \) when \( x = \frac{8x_1}{27} \) is:
\[ \boxed{675} \]