To determine which rectangular equation corresponds to the given parametric equations [tex]\( x = t^{\frac{1}{2}} \)[/tex] and [tex]\( y = 4t \)[/tex], we will follow a series of steps to eliminate the parameter [tex]\( t \)[/tex] and express [tex]\( y \)[/tex] directly in terms of [tex]\( x \)[/tex].
1. Express [tex]\( t \)[/tex] in terms of [tex]\( x \)[/tex]:
Given [tex]\( x = t^{\frac{1}{2}} \)[/tex], we can solve for [tex]\( t \)[/tex]:
[tex]\[
x = t^{\frac{1}{2}}
\][/tex]
Squaring both sides to solve for [tex]\( t \)[/tex], we get:
[tex]\[
x^2 = (t^{\frac{1}{2}})^2
\][/tex]
[tex]\[
x^2 = t
\][/tex]
2. Substitute [tex]\( t \)[/tex] into the equation for [tex]\( y \)[/tex]:
Using the equation [tex]\( y = 4t \)[/tex], we substitute [tex]\( t \)[/tex] with [tex]\( x^2 \)[/tex]:
[tex]\[
y = 4t
\][/tex]
[tex]\[
y = 4(x^2)
\][/tex]
[tex]\[
y = 4x^2
\][/tex]
Therefore, the rectangular equation that represents the given parametric equations is:
[tex]\[
y = 4x^2
\][/tex]
Since the parametric equation [tex]\( x = t^{\frac{1}{2}} \)[/tex] implies that [tex]\( x \geq 0 \)[/tex] (because the square root of a non-negative number is non-negative), this condition must be carried over to the rectangular equation. Thus, for the solution, we have:
[tex]\[
y = 4x^2, \quad \text{for } x \geq 0
\][/tex]
Hence, the correct answer is [tex]\( \boxed{y = 4x^2} \)[/tex] for [tex]\( x \geq 0 \)[/tex].
