Let `f(x,y) = {(x,y): y^(2) le 4x,0 le x le lambda}` and `s(lambda)` is area such that `(S(lambda))/(S(4)) = (2)/(5)`. Find the value of `lambda`.

Options

A. `4((4)/(25))^((1)/(3))`
B. `2((2)/(5))^((1)/(3))`
C. `2((4)/(25))^((1)/(3))`
D. `4((2)/(5))^((1)/(3))`

Answer

To solve the problem, we need to find the value of \( \lambda \) such that the area \( S(\lambda) \) is related to the area \( S(4) \) by the equation:

\[ \frac{S(\lambda)}{S(4)} = \frac{2}{5} \]

### Step 1: Determine the area \( S(\lambda) \)

The area \( S(\lambda) \) is defined by the region bounded by the parabola \( y^2 = 4x \) and the lines \( x = 0 \) and \( x = \lambda \).

The equation of the parabola can be rewritten to express \( y \):

\[ y = \pm 2\sqrt{x} \]

Thus, the area can be calculated as:

\[ S(\lambda) = 2 \int_0^{\lambda} 2\sqrt{x} \, dx \]

### Step 2: Calculate the integral

Calculating the integral:

\[ S(\lambda) = 2 \int_0^{\lambda} 2\sqrt{x} \, dx = 4 \int_0^{\lambda} \sqrt{x} \, dx \]

The integral of \( \sqrt{x} \) is:

\[ \int \sqrt{x} \, dx = \frac{2}{3} x^{3/2} \]

Thus, we have:

\[ S(\lambda) = 4 \left[ \frac{2}{3} x^{3/2} \right]_0^{\lambda} = 4 \cdot \frac{2}{3} \lambda^{3/2} = \frac{8}{3} \lambda^{3/2} \]

### Step 3: Determine the area \( S(4) \)

Now, we compute \( S(4) \):

\[ S(4) = 4 \int_0^{4} \sqrt{x} \, dx = 4 \cdot \frac{2}{3} [x^{3/2}]_0^{4} = 4 \cdot \frac{2}{3} \cdot (4^{3/2}) = 4 \cdot \frac{2}{3} \cdot 8 = \frac{64}{3} \]

### Step 4: Set up the ratio

Now we can set up the equation based on the given ratio:

\[ \frac{S(\lambda)}{S(4)} = \frac{2}{5} \]

Substituting the values we found:

\[ \frac{\frac{8}{3} \lambda^{3/2}}{\frac{64}{3}} = \frac{2}{5} \]

This simplifies to:

\[ \frac{8 \lambda^{3/2}}{64} = \frac{2}{5} \]

### Step 5: Solve for \( \lambda^{3/2} \)

Cross-multiplying gives:

\[ 8 \lambda^{3/2} \cdot 5 = 2 \cdot 64 \]

This simplifies to:

\[ 40 \lambda^{3/2} = 128 \]

Dividing both sides by 40:

\[ \lambda^{3/2} = \frac{128}{40} = \frac{16}{5} \]

### Step 6: Solve for \( \lambda \)

Now, we need to solve for \( \lambda \):

\[ \lambda = \left( \frac{16}{5} \right)^{\frac{2}{3}} \]

### Step 7: Simplify \( \lambda \)

We can express \( 16 \) as \( 2^4 \):

\[ \lambda = \left( \frac{2^4}{5} \right)^{\frac{2}{3}} = \frac{2^{\frac{8}{3}}}{5^{\frac{2}{3}}} \]

Thus, the final value of \( \lambda \) is:

\[ \lambda = 4 \cdot \left( \frac{2}{5} \right)^{\frac{2}{3}} \]

### Final Answer

\[ \lambda = 4 \cdot \left( \frac{2^2}{5^2} \right)^{\frac{1}{3}} = 4 \cdot \frac{4}{25^{\frac{1}{3}}} \]