To find a Cartesian equation of the given parametric equations $x(t)$ and $y(t)$ in the form $x = f(y)$, we need to eliminate the parameter $t$. Here are the parametric equations:

$x(t) = 5t^2$ $y(t) = -10 + 4t$

Step 1: Solve for $t$ in terms of $y$

From the second equation $y(t) = -10 + 4t$, solve for $t$:

$y + 10 = 4t$ $t = \frac{y + 10}{4}$

Step 2: Substitute $t$ into the $x(t)$ equation

Now, substitute $t = \frac{y + 10}{4}$ into the first equation $x(t) = 5t^2$:

$x = 5 \left( \frac{y + 10}{4} \right)^2$

Step 3: Simplify the expression

Simplify the equation:

$x = 5 \cdot \frac{(y + 10)^2}{16}$ $x = \frac{5(y + 10)^2}{16}$

Final Cartesian equation:

The Cartesian equation in the form $x = f(y)$ is:

$x = \frac{5(y + 10)^2}{16}$

Would you like further details on any of these steps?

Here are some related questions:

1. How do parametric equations differ from Cartesian equations?
2. What are common methods for eliminating parameters in parametric equations?
3. How do you solve more complex parametric equations?
4. What real-world applications use parametric equations?
5. What is the geometric interpretation of the given parametric equations?

Tip: Always isolate one variable first when eliminating parameters—it simplifies the process.