# Parametric equations for a loop

• ILoveBaseball
In summary: Eq} for "x" indicates t_{1} = -t_{2} and \mathsf{Eq} for "y" uses the next hint. \frac { t_{1}^{3} \ - \ t_{2}^{3} } { t_{1} \ - \ t_{2} } \ = \ t_{1}^{2} \ + \ t_{1}t_{2} \ + \ t_{2}^{2} (\t_{1} \ \ne \ t_{2})\Here are some HINTS:a): t_{1} = t_{2} if \t_{1
ILoveBaseball
The following parametric equations trace out a loop
$$x = 8 - 3/2t^2$$
$$y = -3/6t^3+3t+1$$

1.) Find the $$t$$ values at which the curve intersects itself.
wouldn't i just have to solve for t for one of the equaltion to find t? also, can you find the intersects using a TI-83 plus to check your answer?

$$8 - 3/2t^2=0$$
$$3/2t^2=8$$
$$t^2=16/3$$
$$t = \pm\sqrt(16/3)$$ which is wrong

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You can use the ti83 to find them.
Put it in polar mode, enter your equations, and from there find intersects like you normally would.

there is no "intersect" function when i set it in parametic mode. How would i find the T values manually?

Well what you did was find the y intercepts of the graph. I looked in my calc book and can't find anything on intersections of parametric curves.

so how would i find the $$t$$ values?

ILoveBaseball said:
The following parametric equations trace out a loop
$$x = 8 - 3/2t^2$$
$$y = -3/6t^3+3t+1$$

1.) Find the $$t$$ values at which the curve intersects itself.
wouldn't i just have to solve for t for one of the equaltion to find t? also, can you find the intersects using a TI-83 plus to check your answer?

$$8 - 3/2t^2=0$$
$$3/2t^2=8$$
$$t^2=16/3$$
$$t = \pm\sqrt(16/3)$$ which is wrong
The curve will intersect itself when the same "x" and "y" values are produced by 2 different "t" values, say "t1" and "t2". Thus:

$$x \ = \ 8 \ - \ (3/2) t_{1}^{2} \ = \ 8 \ - \ (3/2)t_{2}^{2}$$
$$y \ = \ (-3/6) t_{1}^{3} \ + \ 3 t_{1} + 1 \ = \ (-3/6) t_{2}^{3} \ + \ 3 t_{2} + 1$$

Solve for DIFFERENT values of "t1" and "t2" to find the crossing point.
(Solve the equations simultaneously. Both equations must be satisfied.)

~~

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The curve will intersect itself when the same "x" and "y" values are produced by 2 different "t" values, say "t1" and "t2". Thus:

$$x \ = \ 8 \ - \ (3/2) t_{1}^{2} \ = \ 8 \ - \ (3/2)t_{2}^{2}$$

$$y \ = \ (-3/6) t_{1}^{3} \ + \ 3 t_{1} + 1 \ = \ (-3/6) t_{2}^{3} \ + \ 3 t_{2} + 1$$

Solve for DIFFERENT values of "t1" and "t2" to find the crossing point.
(Solve the equations simultaneously. Both equations must be satisfied.)

Here are some HINTS:

$$\color{blue} a): \ \ \ \ \mathsf{ Eq \ for \ "x" \ indicates \ t_{1} = -t_{2} \ \ \ \ \ \ \ Eq \ for \ "y" \ uses \ next \ hint }$$

$$\color{blue} b): \ \ \ \ \frac { t_{1}^{3} \ - \ t_{2}^{3} } { t_{1} \ - \ t_{2} } \ = \ t_{1}^{2} \ + \ t_{1}t_{2} \ + \ t_{2}^{2} \ \ \ \ \ \ \ ( t_{1} \ \ne \ t_{2} )$$

$$\color{blue} c): \ \ \ \ (Answers) \ \longrightarrow \ \ (t_{1} \ = \ \sqrt{6}) \ \ and \ \ (t_{2} \ =\ -\sqrt{6})$$

~~

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## 1. What are parametric equations for a loop?

Parametric equations for a loop are a set of equations that describe the motion of a particle or object as it moves along a loop or closed curve. These equations typically involve a variable parameter, such as time, that allows us to trace the path of the object as it moves through the loop.

## 2. How are parametric equations for a loop different from regular equations?

Parametric equations for a loop differ from regular equations in that they use a parameter to represent the independent variable, rather than a traditional variable like x or y. This allows for a more dynamic and flexible representation of the object's motion along the loop.

## 3. What are the advantages of using parametric equations for a loop?

There are several advantages to using parametric equations for a loop. One advantage is that they can provide a more precise description of the object's motion, especially in complex loops. They also allow for the use of more advanced mathematical techniques, such as calculus, to analyze the motion and make predictions.

## 4. How are parametric equations for a loop used in real-world applications?

Parametric equations for a loop have many practical applications, such as in physics and engineering. They can be used to model the motion of objects in circular or curved paths, such as roller coasters or satellites in orbit. They can also be used in computer graphics to create realistic animations of moving objects.

## 5. Are there any limitations to using parametric equations for a loop?

While parametric equations for a loop can be useful in many situations, they do have some limitations. One limitation is that they can be more complex and difficult to work with compared to regular equations. Additionally, they may not be suitable for describing certain types of motion, such as non-closed curves or irregular shapes.

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