Problem 11 - Algebra and Trigonometry

Question

\(3-24=A\) pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular coordinate equation for the curve by eliminating the parameter. $$ x=4 t^{2}, \quad y=8 t^{3} $$

Step-by-Step Solution

Answer

The rectangular coordinate equation is \( y = x^{3/2} \), representing a parabola opening upwards.

Step 1: Understand the Parametric Equations

We have the parametric equations given as \( x = 4t^2 \) and \( y = 8t^3 \). Observe that these equations use a parameter \( t \) to define the relationship between \( x \) and \( y \).

Step 2: Express Parameter \( t \) in Terms of \( x \)

To eliminate the parameter \( t \), solve for \( t \) in terms of \( x \). Since \( x = 4t^2 \), divide both sides by 4 to get \( t^2 = \frac{x}{4} \). Thus, \( t = \sqrt{\frac{x}{4}} \).

Step 3: Substitute \( t \) in Terms of \( x \) into \( y \) Equation

Now substitute \( t = \sqrt{\frac{x}{4}} \) into the equation \( y = 8t^3 \). It gives \( y = 8 (\sqrt{\frac{x}{4}})^3 = 8 \left( \frac{x}{4} \right)^{3/2} \). Simplify the expression to obtain \( y = \frac{8}{8} \cdot (x^{3/2}) = x^{3/2} \).

Step 4: Rectangular Coordinate Equation

The rectangular coordinate equation for the curve is \( y = x^{3/2} \). This expresses \( y \) directly in terms of \( x \), with the parameter \( t \) eliminated.

Step 5: Sketch the Curve

To sketch the curve of the equation \( y = x^{3/2} \), note that it represents a portion of a parabola. As \( x \) increases from 0, \( y \) increases at an increasing rate because of the exponent \( 3/2 \). The curve opens upwards from the origin, and we only consider non-negative \( x \) since \( x = 4t^2 \) for \( t \geq 0 \).

Key Concepts

Rectangular Coordinate Equation

In mathematics, a rectangular coordinate equation attempts to express a relationship between two variables. This equation is much like a standard Cartesian coordinate system, which uses

• x
• y

to represent any point in a plane.
Rectangular coordinates are

• simple
• intuitive

making them preferred when trying to represent curves without using parameters.
In this particular exercise, we have the parametric equations:

• \( x = 4t^2 \)
• \( y = 8t^3 \)

We arrive at the rectangular coordinate equation after a series of transformations to eliminate the parameter \( t \), replacing it with a direct relationship between \( x \) and \( y \). This equation is purposefully written to share the natural geometry of the shape or curve, which in this case becomes

• \( y = x^{3/2} \).

This gives us a clear geometric representation without needing any intermediaries like the parameter \( t \).!

Parameter Elimination

Parameter elimination is the process of removing the third variable (the parameter) from parametric equations to express the relationship directly between \( x \) and \( y \). This is often done when one wants to understand the geometry or shape implied by the equations at hand without referring explicitly to the parameter.
In the context of this task, we began with:

• \( x = 4t^2 \)
• \( y = 8t^3 \)

Our goal was to remove \( t \) to discover the rectangular coordinate equation.
**Steps Involved:**

• First, solve for \( t \) in any one of the given parametric equations. For example, from \( x = 4t^2 \), we find \( t^2 = \frac{x}{4} \), giving us \( t = \sqrt{\frac{x}{4}} \).
• We then substitute this expression back into the other parametric equation for y. Using \( y = 8t^3 \), replace \( t \) with \( \sqrt{\frac{x}{4}} \), which results in \( y = 8\left(\sqrt{\frac{x}{4}}\right)^3 \).

After simplifying, \( y = x^{3/2} \) becomes the desired rectangular equation.
Proper parameter elimination helps in converting equations into a more geometrically intuitive format, making it easier to sketch and understand the associated curve.

Parametric to Cartesian Conversion

Moving from parametric equations to a Cartesian form is known as parametric to Cartesian conversion. This transformation helps you understand relationships directly between variables \( x \) and \( y \) in a Cartesian coordinate system, portraying the curve's shape in a more straightforward way.
**Steps for Conversion:**1. **Identify the Parametric Equations**: These are given as \( x = 4t^2 \) and \( y = 8t^3 \).2. **Express \( t \) in terms of \( x \)**: Start by rearranging \( x = 4t^2 \) to solve for \( t \) as \( t = \sqrt{\frac{x}{4}} \).3. **Substitute \( t \) in the \( y \) equation**: Using \( y = 8t^3 \), you add the expression for \( t \), getting \( y = 8\left(\frac{x}{4}\right)^{3/2} \).4. **Simplify the Equation**: This lands you the simplified Cartesian equation \( y = x^{3/2} \).This conversion tells us how \( x \) and \( y \) correlate, neatly sidelining the parameter. With your new Cartesian equation, graphing becomes simpler, focusing purely on how one coordinate affects the other.