Question

Rewrite the following parametric equations by solving for y.

"x left parenthesis t right parenthesis equals e superscript negative t baseline y left parenthesis t right parenthesis equals 3 e superscript 2 t baseline"
(1 point)
Responses

y equals one-third x squared comma x is greater than 0
Image with alt text: y equals one-third x squared comma x is greater than 0

y equals start fraction 3 over x squared end fraction comma x is greater than 0
Image with alt text: y equals start fraction 3 over x squared end fraction comma x is greater than 0

y equals 3 e superscript t baseline comma, x is greater than 0
Image with alt text: y equals 3 e superscript t baseline comma, x is greater than 0

y equals 4 e superscript t baseline comma, x is greater than 0

Answer 1

To rewrite the parametric equations \( x(t) = e^{-t} \) and \( y(t) = 3 e^{2t} \) by solving for \( y \) in terms of \( x \), we can follow these steps:

1. Start with the equation for \( x(t) \):
\[
x = e^{-t}
\]

2. Solve for \( t \):
\[
t = -\ln(x)
\]

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

Thus, we get the equation:
\[
y = \frac{3}{x^2}, \quad x > 0
\]

So, the correct rewritten parametric equations are:
\[
y = \frac{3}{x^2}, \quad x > 0
\]

The corresponding response is:
- **y equals \(\frac{3}{x^2}\), x is greater than 0**.