/*! 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 24 Find the points at which the fol... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 24

Find the points at which the following planes intersect the coordinate axes and find equations of the lines where the planes intersect the coordinate planes. Sketch a graph of the plane. $$12 x-9 y+4 z+72=0$$

Short Answer

Expert verified
Answer: The intersection points with the coordinate axes are $$(-6, 0, 0)$$, $$(0, 8, 0)$$, and $$(0, 0, -18)$$. The equations of the lines where the plane intersects the coordinate planes are $$y = \frac{4}{3}x + 8$$ (xy-plane), $$z = \frac{9}{4}y - 18$$ (yz-plane), and $$z = -3x - 18$$ (xz-plane).

Step by step solution

01

Find the points at which the plane intersects the coordinate axes

To find the points at which the plane intersects the coordinate axes, set two of the three coordinates equal to zero and solve for the remaining one. 1. Intersection with x-axis (y = 0, z = 0): $$12x + 0 - 0 + 72 = 0$$ Solve for x: $$12x = -72$$ $$x = -6$$ Intersection point: $$(-6, 0, 0)$$ 2. Intersection with y-axis (x = 0, z = 0): $$0 - 9y + 0 + 72 = 0$$ Solve for y: $$-9y = -72$$ $$y = 8$$ Intersection point: $$(0, 8, 0)$$ 3. Intersection with z-axis (x = 0, y = 0): $$0 + 0 + 4z + 72 = 0$$ Solve for z: $$4z = -72$$ $$z = -18$$ Intersection point: $$(0, 0, -18)$$
02

Find the equations of the lines where the plane intersects the coordinate planes

To find the equations of the lines where the plane intersects the coordinate planes, set the coordinate associated with the plane equal to zero and solve for the other two coordinates. 1. Intersection with xy-plane (z = 0): $$12x - 9y + 72 = 0$$ Rearrange the equation to solve for y: $$y = \frac{4}{3}x + 8$$ This is the equation of the line on the xy-plane. 2. Intersection with yz-plane (x = 0): $$-9y + 4z + 72 = 0$$ Rearrange the equation to solve for z: $$z = \frac{9}{4}y - 18$$ This is the equation of the line on the yz-plane. 3. Intersection with xz-plane (y = 0): $$12x + 4z + 72 = 0$$ Rearrange the equation to solve for z: $$z = -3x - 18$$ This is the equation of the line on the xz-plane.
03

Sketch a graph of the plane

Now that we know the intersection points with the coordinate axes and the equations of the lines where the plane intersects the coordinate planes, we can plot these points and lines on a 3D coordinate space. 1. Plot the points at which the plane intersects the coordinate axes: $$(-6, 0, 0)$$, $$(0, 8, 0)$$, and $$(0, 0, -18)$$. 2. Draw the lines connecting the points to show the plane's intersection with the coordinate planes, using the line equations obtained in step 2: $$y = \frac{4}{3}x + 8$$, $$z = \frac{9}{4}y - 18$$, and $$z = -3x - 18$$. 3. Fill in the plane that contains these lines and points to complete the sketch of the plane. This concludes the breakdown of the exercise.

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

Describe the set of all points at which all three planes \(x+3 z=3, y+4 z=6,\) and \(x+y+6 z=9\) intersect.

Let \(x, y,\) and \(z\) be nonnegative numbers with \(x+y+z=200\). a. Find the values of \(x, y,\) and \(z\) that minimize \(x^{2}+y^{2}+z^{2}\). b. Find the values of \(x, y,\) and \(z\) that minimize \(\sqrt{x^{2}+y^{2}+z^{2}}\). c. Find the values of \(x, y,\) and \(z\) that maximize \(x y z\). d. Find the values of \(x, y,\) and \(z\) that maximize \(x^{2} y^{2} z^{2}\).

Evaluate the following limits. $$\lim _{(x, y) \rightarrow(0,2)}(2 x y)^{x y}$$

Limits at (0,0) may be easier to evaluate by converting to polar coordinates. Remember that the same limit must be obtained as \(r \rightarrow 0\) along all paths to (0,0) Evaluate the following limits or state that they do not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{x-y}{\sqrt{x^{2}+y^{2}}}$$

Suppose \(n\) houses are located at the distinct points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right) .\) A power substation must be located at a point such that the sum of the squares of the distances between the houses and the substation is minimized. a. Find the optimal location of the substation in the case that \(n=3\) and the houses are located at \((0,0),(2,0),\) and (1,1) b. Find the optimal location of the substation in the case that \(n=3\) and the houses are located at distinct points \(\left(x_{1}, y_{1}\right)\) \(\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) c. Find the optimal location of the substation in the general case of \(n\) houses located at distinct points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots\) \(\left(x_{n}, y_{n}\right)\) d. You might argue that the locations found in parts (a), (b), and (c) are not optimal because they result from minimizing the sum of the squares of the distances, not the sum of the distances themselves. Use the locations in part (a) and write the function that gives the sum of the distances. Note that minimizing this function is much more difficult than in part (a). Then use a graphing utility to determine whether the optimal location is the same in the two cases. (Also see Exercise 75 about Steiner's problem.)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

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.