/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 82 Which of the following parametri... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 82

Which of the following parametric equations describe the same curve? a. \(x=2 t^{2}, y=4+t ;-4 \leq t \leq 4\) b. \(x=2 t^{4}, y=4+t^{2} ;-2 \leq t \leq 2\) c. \(x=2 t^{2 / 3}, y=4+t^{1 / 3} ;-64 \leq t \leq 64\)

Short Answer

Expert verified
a. \(x = 2t^2\), \(y = 4 + t\), with \(-4 \leq t \leq 4\) b. \(x = 2t^4\), \(y = 4 + t^2\), with \(-2 \leq t \leq 2\) c. \(x = 2t^{\frac{2}{3}}\), \(y = 4 + t^{\frac{1}{3}}\), with \(-64 \leq t \leq 64\) Answer: None of the given parametric equations describe the same curve.

Step by step solution

01

Identify the functional relationships

In order to determine whether the given parametric equations describe the same curve, we need to compare the functional relationships between \(x\) and \(y\) for each set. a. \(x = 2t^2\), \(y = 4 + t\) b. \(x = 2t^4\), \(y = 4 + t^2\) c. \(x = 2t^{\frac{2}{3}}\), \(y = 4 + t^{\frac{1}{3}}\)
02

Eliminate parameter t

Now, we will eliminate the parameter \(t\) in each set of equations and rewrite the equations in terms of \(x\) and \(y\) only. a. Solve for \(t\) in the first equation: \(t = \pm\sqrt{\frac{x}{2}}\). Substitute this into the second equation: \(y = 4 \pm \sqrt{\frac{x}{2}}\). b. Solve for \(t\) in the first equation: \(t = \pm\sqrt[4]{\frac{x}{2}}\). Substitute this into the second equation: \(y = 4 + \frac{x}{2}\). c. Solve for \(t\) in the first equation: \(t = (\frac{x}{2})^{\frac{3}{2}}\). Substitute this into the second equation: \(y = 4 + (\frac{x}{2})^{\frac{1}{2}}\).
03

Compare functional relationships

Compare the relationships between \(x\) and \(y\) in the rewritten equations. a. \(y = 4 \pm \sqrt{\frac{x}{2}}\) b. \(y = 4 + \frac{x}{2}\) c. \(y = 4 + (\frac{x}{2})^{\frac{1}{2}}\) We can see that none of these relationships between \(x\) and \(y\) are the same, so none of the parametric equations describe the same curve.
04

Consider the range of parameter t

Even though the curves are different in their functional relationships, we should also check the range of \(t\) for each set of parametric equations, as this could lead to different curve segments. a. \(-4 \leq t \leq 4\) b. \(-2 \leq t \leq 2\) c. \(-64 \leq t \leq 64\) These do not have the same range; thus, even if the equations had the same relationships between \(x\) and \(y\), the curves represented by the equations would be different segments.
05

Conclusion

None of the given parametric equations describe the same curve, as neither the functional relationships nor the range of parameter \(t\) are the same for any pair of equations.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique. The upper half of the parabola \(x=y^{2}\), originating at \((0,0)\)

Consider the following Lissajous curves. Graph the curve and estimate the coordinates of the points on the curve at which there is (a) a horizontal tangent line and (b) a vertical tangent line. (See the Guided Project Parametric art for more on Lissajous curves.) $$\begin{aligned}&x=\sin 2 t, y=2 \sin t\\\&0 \leq t \leq 2 \pi\end{aligned}$$

The Lam茅 curve described by \(\left|\frac{x}{a}\right|^{n}+\left|\frac{y}{b}\right|^{n}=1,\) where \(a, b,\) and \(n\) are positive real numbers, is a generalization of an ellipse. a. Express this equation in parametric form (four pairs of equations are needed). b. Graph the curve for \(a=4\) and \(b=2,\) for various values of \(n\) c. Describe how the curves change as \(n\) increases.

Find an equation of the line tangent to the hyperbola \(x^{2} / a^{2}-y^{2} / b^{2}=1\) at the point \(\left(x_{0}, y_{0}\right)\)

Find real numbers a and b such that equations \(A\) and \(B\) describe the same curve. \(A: x=t+t^{3}, y=3+t^{2} ;-2 \leq t \leq 2\) \(B: x=t^{1 / 3}+t, y=3+t^{2 / 3} ; a \leq t \leq b\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.