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x=4 t² , y=8 t³ A pair of parametric equations is given. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter.

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Final answer:

To eliminate the parameter t in the given pair of parametric equations x=4t² and y=8t³, we can express t in terms of x and substitute it into the equation for y. Simplifying further, we obtain the rectangular-coordinate equation y = sqrt(x)³.

Step-by-step explanation:

To eliminate the parameter t, we can express t in terms of x and substitute it into the equation for y. From the equation x = 4t², we can solve for t:

t = sqrt(x/4)

Substituting this value of t into the equation for y = 8t³, we get:

y = 8(sqrt(x/4))³

Simplifying further, we have:

y = 8(sqrt(x/4))³

y = 8(sqrt(x)/2)³

y = 8(sqrt(x)³/8)

y = sqrt(x)³

Therefore, the rectangular-coordinate equation for the curve is y = sqrt(x)³.

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