If `y=propx^(4)`
When`x=1000, y=y_(1)`
then find the value of x so that value of y can be `2y_(1)` ,

Options

A. 2000
B. 1190
C. 1410
D. 4000

Answer

To solve the problem step by step, we will follow the logic presented in the video transcript. The problem states that \( y \) is directly proportional to \( x^4 \) and we need to find the value of \( x \) such that \( y = 2y_1 \) when \( x = 1000 \) gives \( y = y_1 \).

### Step 1: Establish the relationship
Since \( y \) is directly proportional to \( x^4 \), we can express this relationship mathematically as:
\[ y = kx^4 \]
where \( k \) is the constant of proportionality.

### Step 2: Find the constant \( k \)
When \( x = 1000 \), we have:
\[ y_1 = k(1000)^4 \]
Calculating \( (1000)^4 \):
\[ (1000)^4 = 10^{12} \]
Thus, we can express \( k \) as:
\[ k = \frac{y_1}{10^{12}} \]

### Step 3: Set up the equation for \( y = 2y_1 \)
Now, we need to find \( x \) such that:
\[ y = 2y_1 \]
Substituting the expression for \( y \):
\[ 2y_1 = kx^4 \]
Substituting the value of \( k \):
\[ 2y_1 = \left(\frac{y_1}{10^{12}}\right)x^4 \]

### Step 4: Simplify the equation
Dividing both sides by \( y_1 \) (assuming \( y_1 \neq 0 \)):
\[ 2 = \frac{x^4}{10^{12}} \]
Multiplying both sides by \( 10^{12} \):
\[ x^4 = 2 \times 10^{12} \]

### Step 5: Solve for \( x \)
Taking the fourth root of both sides:
\[ x = \sqrt[4]{2 \times 10^{12}} \]
This can be simplified as:
\[ x = \sqrt[4]{2} \times \sqrt[4]{10^{12}} = \sqrt[4]{2} \times 10^3 \]
Calculating \( \sqrt[4]{10^{12}} \):
\[ \sqrt[4]{10^{12}} = 10^{3} = 1000 \]
Thus, we have:
\[ x = 1000 \times \sqrt[4]{2} \]

### Step 6: Calculate \( \sqrt[4]{2} \)
Using the approximation \( \sqrt[4]{2} \approx 1.189 \):
\[ x \approx 1000 \times 1.189 \approx 1189 \]

### Step 7: Final answer
Rounding \( 1189 \) gives us approximately \( 1190 \). Therefore, the value of \( x \) such that \( y = 2y_1 \) is:
\[ \boxed{1190} \]