Rewrite the following parametric equations in rectangular form.

Question

x equals e superscript 3 t baseline and y equals e superscript negative t baseline
(1 point)
Responses

y equals Start Fraction 3 over x End Fraction,x greater than 0
Image with alt text: y equals Start Fraction 3 over x End Fraction , Image with alt text: x greater than 0

x equals Start Fraction 1 over y cubed End Fraction,y greater than 0
Image with alt text: x equals Start Fraction 1 over y cubed End Fraction , Image with alt text: y greater than 0

x equals negative y cubed,y less than 0
Image with alt text: x equals negative y cubed , Image with alt text: y less than 0

y equals Start Fraction 1 over x cubed End Fraction,x greater than 0

Answer 1

To convert the parametric equations \( x = e^{3t} \) and \( y = e^{-t} \) into rectangular form, we can express \( t \) in terms of \( x \) and substitute it into the equation for \( y \):

1. Start with the equation for \( x \): \[ x = e^{3t} \] Taking the natural logarithm of both sides, we get: \[ 3t = \ln(x) \quad \Rightarrow \quad t = \frac{\ln(x)}{3} \]

2. Now substitute \( t \) into the equation for \( y \): \[ y = e^{-t} = e^{-\frac{\ln(x)}{3}} = x^{-\frac{1}{3}} = \frac{1}{x^{1/3}} \]

Thus, the rectangular form of the parametric equations is: \[ y = \frac{1}{x^{1/3}} \quad \text{for } x > 0 \]

So, the correct response is:

• \( y = \frac{1}{x^{3}} , x > 0 \) (It seems like the context of \( y = \frac{1}{x^{1/3}} \) was misinterpreted; clarify the exponent regarding whether it should be \( 3 \) or \( \frac{1}{3} \) based on your note as there is an inconsistency.)