/*! 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 91 Find the points (if they exist) ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 91

Find the points (if they exist) at which the following planes and curves intersect. $$8 x+y+z=60 ; \quad \mathbf{r}(t)=\left\langle t, t^{2}, 3 t^{2}\right\rangle, \text { for }-\infty

Short Answer

Expert verified
Question: Find the points at which the plane \(8x + y + z = 60\) intersects the curve given by the vector function \(\mathbf{r}(t) = \langle t, t^2, 3t^2 \rangle\). Answer: The intersection points are: $$P_1 = \left( \frac{-8 + \sqrt{1184}}{8}, \left(\frac{-8 + \sqrt{1184}}{8}\right)^2, 3\left(\frac{-8 + \sqrt{1184}}{8}\right)^2 \right)$$ $$P_2 = \left( \frac{-8 - \sqrt{1184}}{8}, \left(\frac{-8 - \sqrt{1184}}{8}\right)^2, 3\left(\frac{-8 - \sqrt{1184}}{8}\right)^2 \right)$$

Step by step solution

01

Identify the equation of the plane

The given equation of the plane is: $$8x + y + z = 60$$
02

Parameterize the curve

The given parameterized curve is: $$\mathbf{r}(t) = \langle t, t^2, 3t^2 \rangle$$ This means that the function is defined in terms of the parameter \(t\) as follows: $$x=t, \quad y=t^2, \quad z=3t^2$$
03

Substitute the parameterized curve into the plane's equation

To find the intersection points, substitute the parameterized curve into the plane's equation: $$8(t) + (t^2) + (3t^2) = 60$$
04

Solve for t

Now, solve the equation to find the value(s) of \(t\): $$8t + t^2 + 3t^2 = 60$$ $$4t^2 + 8t - 60 = 0$$ This is a quadratic equation. To find the roots, we can use the quadratic formula: $$t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where \(a = 4, b = 8,\) and \(c = -60\). Substitute the values of \(a, b, c\) to find \(t\): $$t = \frac{-8 \pm \sqrt{8^2 - 4(4)(-60)}}{2(4)}$$ $$t = \frac{-8 \pm \sqrt{1184}}{8}$$ Now, we have two values of \(t\): $$t_1 = \frac{-8 + \sqrt{1184}}{8}$$ $$t_2 = \frac{-8 - \sqrt{1184}}{8}$$
05

Find the intersection points

Use the parameterized curve to find the intersection points by substituting the values of \(t\): For \(t_1 = \frac{-8 + \sqrt{1184}}{8}\): $$x_1 = t_1, \quad y_1 = t_1^2, \quad z_1 = 3t_1^2$$ For \(t_2 = \frac{-8 - \sqrt{1184}}{8}\): $$x_2 = t_2, \quad y_2 = t_2^2, \quad z_2 = 3t_2^2$$ The intersection points are: $$P_1 = \left( \frac{-8 + \sqrt{1184}}{8}, \left(\frac{-8 + \sqrt{1184}}{8}\right)^2, 3\left(\frac{-8 + \sqrt{1184}}{8}\right)^2 \right)$$ $$P_2 = \left( \frac{-8 - \sqrt{1184}}{8}, \left(\frac{-8 - \sqrt{1184}}{8}\right)^2, 3\left(\frac{-8 - \sqrt{1184}}{8}\right)^2 \right)$$ These are the points of intersection between the given plane and curve.

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

Identify and briefly describe the surfaces defined by the following equations. $$x^{2} / 4+y^{2}-2 x-10 y-z^{2}+41=0$$

Absolute maximum and minimum values Find the absolute maximum and minimum values of the following functions over the given regions \(R\). Use Lagrange multipliers to check for extreme points on the boundary. $$f(x, y)=2 x^{2}+y^{2}+2 x-3 y ; R=\left\\{(x, y): x^{2}+y^{2} \leq 1\right\\}$$

Find an equation of the plane passing through (0,-2,4) that is orthogonal to the planes \(2 x+5 y-3 z=0\) and \(-x+5 y+2 z=8\)

Find the points (if they exist) at which the following planes and curves intersect. $$y=2 x+1 ; \quad \mathbf{r}(t)=\langle 10 \cos t, 2 \sin t, 1\rangle, \text { for } 0 \leq t \leq 2 \pi$$

Maximizing utility functions Find the values of \(\ell\) and \(g\) with \(\ell \geq 0\) and \(g \geq 0\) that maximize the following utility functions subject to the given constraints. Give the value of the utility function at the optimal point. $$U=f(\ell, g)=8 \ell^{4 / 5} g^{1 / 5} \text { subject to } 10 \ell+8 g=40$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

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.