Problem 120 - Algebra and Trigonometry

Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find a rectangular equation of the plane curve with parametric equations \(x(t)=t+5\) and \(y(t)=\sqrt{t}\) for \(t \geq 0\).

Step-by-Step Solution

Answer

The rectangular equation is \( y = \sqrt{x - 5} \).

Step 1 - Express t in terms of x

Given the parametric equation for x: \( x = t + 5 \)Solve for \( t \) to find: \( t = x - 5 \)

Step 2 - Substitute t in y equation

Next, using the parametric equation for y: \( y = \sqrt{t} \)Substitute \( t \) from the previous step: \( y = \sqrt{x - 5} \)

Step 3 - Combine the equations

Now we have the relation between \( x \) and \(-\) \( y \). The rectangular equation of the plane curve is: \( y = \sqrt{x - 5} \)

Key Concepts

Rectangular Equation

When dealing with parametric equations, our goal is often to transform these equations into a single, non-parametric equation. This form is known as the *rectangular equation*. In the given exercise, we start with two parametric equations: one for x and one for y. To find the rectangular equation, we first solve one of the parametric equations for the parameter (in this case, t). Then, we substitute this expression into the other parametric equation. In simpler terms, we remove the parameter t from both equations, leaving us with a direct relationship between x and y. Transforming parametric equations into a rectangular equation can simplify the curve and makes it easier to graph and understand.

Parametric Curves

A *parametric curve* is defined by a set of parametric equations. These equations use parameters, like t, to express both the x and y coordinates. For the given problem, x(t) = t + 5 and y(t) = \( \sqrt{t} \) describe the curve's behavior in the plane. Parametric curves are useful for representing complex behaviors that are difficult to describe using a single equation. They are particularly handy in physics and engineering because they provide a more detailed way to describe an object's position over time. To get a clearer picture, imagine the curve as a path traced by a particle moving through space and the parameter t represents time.

Algebra

Algebra plays a crucial role in transforming and simplifying equations, including parametric equations. In the provided solution, algebraic manipulation was used to isolate the parameter t from one equation. Specifically, from x(t) = t + 5, we get t = x - 5. This expression is then substituted in the equation for y. Next, substituting t into y(t) = \( \sqrt{t} \), we get y = \( \sqrt{x - 5} \). This is the rectangular equation derived from the parametric equations. Basic algebra skills, such as isolating variables and substitution, are vital for converting parametric equations into rectangular forms. By mastering these algebraic techniques, you can easily handle and manipulate various forms of mathematical equations.