31. At what points does the curve r(t) = ti + (2t - 1²) k inter- sect the paraboloid z = x² + y²?

Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
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31,43 pl

n. Indicate
,r)
f, COS TI)
e coordinate
3.
ons for the
Q(-3, 5, 1)
(u, v, w)
phs
25. x = cos 8t, y sin 81, z =
26. x = cos't, y = sin²t, z = 1
0.8
81
120
27. Show that the curve with parametric equations x = 1 cos 1,
y = f sin f, z = t lies on the cone z² = x² + y2, and use this
fact to help sketch the curve.
28. Show that the curve with parametric equations x = sin 1,
y = cost, z = sin²t is the curve of intersection of the surfaces
z = x² and x² + y² = 1. Use this fact to help sketch the curve.
29. Find three different surfaces that contain the curve
r(t) = 2ti + e'j + e¹¹ k.
21
30. Find three different surfaces that contain the curve
r(t) = t²i+ In tj+ (1/1) k.
31. At what points does the curve r(t) = ti + (2t - 1²) k inter-
sect the paraboloid z = x² + y²?
32. At what points does the helix r(t) = (sin t, cost, t) intersect
the sphere x² + y² + z² = 5?
2
33-37 Use a computer to graph the curve with the given vector
equation. Make sure you choose a parameter domain and view-
points that reveal the true nature of the curve.
33. r(t) = (cost sin 2t, sin t sin 2t, cos 2t)
34. r(t) = (te', e, t)
35. r(t) =
(sin 3t cos t, 4t, sin 3t sin t)
36. r(t) = (cos(8 cos t) sin t, sin(8 cos t) sin t, cost)
37. r(t) = (cos 2t, cos 3t, cos 4t)
38. Graph the curve with parametric equations x = sin f,
y = sin 21, z = cos 4t. Explain its shape by graphing its
projections onto the three coordinate planes.
39. Graph the curve with parametric equations
x = (1 + cos 16t) cost
Transcribed Image Text:n. Indicate ,r) f, COS TI) e coordinate 3. ons for the Q(-3, 5, 1) (u, v, w) phs 25. x = cos 8t, y sin 81, z = 26. x = cos't, y = sin²t, z = 1 0.8 81 120 27. Show that the curve with parametric equations x = 1 cos 1, y = f sin f, z = t lies on the cone z² = x² + y2, and use this fact to help sketch the curve. 28. Show that the curve with parametric equations x = sin 1, y = cost, z = sin²t is the curve of intersection of the surfaces z = x² and x² + y² = 1. Use this fact to help sketch the curve. 29. Find three different surfaces that contain the curve r(t) = 2ti + e'j + e¹¹ k. 21 30. Find three different surfaces that contain the curve r(t) = t²i+ In tj+ (1/1) k. 31. At what points does the curve r(t) = ti + (2t - 1²) k inter- sect the paraboloid z = x² + y²? 32. At what points does the helix r(t) = (sin t, cost, t) intersect the sphere x² + y² + z² = 5? 2 33-37 Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and view- points that reveal the true nature of the curve. 33. r(t) = (cost sin 2t, sin t sin 2t, cos 2t) 34. r(t) = (te', e, t) 35. r(t) = (sin 3t cos t, 4t, sin 3t sin t) 36. r(t) = (cos(8 cos t) sin t, sin(8 cos t) sin t, cost) 37. r(t) = (cos 2t, cos 3t, cos 4t) 38. Graph the curve with parametric equations x = sin f, y = sin 21, z = cos 4t. Explain its shape by graphing its projections onto the three coordinate planes. 39. Graph the curve with parametric equations x = (1 + cos 16t) cost
n. Indicate
,r)
f, COS TI)
e coordinate
3.
ons for the
Q(-3, 5, 1)
(u, v, w)
phs
25. x = cos 8t, y sin 81, z =
26. x = cos't, y = sin²t, z = 1
0.8
81
120
27. Show that the curve with parametric equations x = 1 cos 1,
y = f sin f, z = t lies on the cone z² = x² + y2, and use this
fact to help sketch the curve.
28. Show that the curve with parametric equations x = sin 1,
y = cost, z = sin²t is the curve of intersection of the surfaces
z = x² and x² + y² = 1. Use this fact to help sketch the curve.
29. Find three different surfaces that contain the curve
r(t) = 2ti + e'j + e¹¹ k.
21
30. Find three different surfaces that contain the curve
r(t) = t²i+ In tj+ (1/1) k.
31. At what points does the curve r(t) = ti + (2t - 1²) k inter-
sect the paraboloid z = x² + y²?
32. At what points does the helix r(t) = (sin t, cost, t) intersect
the sphere x² + y² + z² = 5?
2
33-37 Use a computer to graph the curve with the given vector
equation. Make sure you choose a parameter domain and view-
points that reveal the true nature of the curve.
33. r(t) = (cost sin 2t, sin t sin 2t, cos 2t)
34. r(t) = (te', e, t)
35. r(t) =
(sin 3t cos t, 4t, sin 3t sin t)
36. r(t) = (cos(8 cos t) sin t, sin(8 cos t) sin t, cost)
37. r(t) = (cos 2t, cos 3t, cos 4t)
38. Graph the curve with parametric equations x = sin f,
y = sin 21, z = cos 4t. Explain its shape by graphing its
projections onto the three coordinate planes.
39. Graph the curve with parametric equations
x = (1 + cos 16t) cost
Transcribed Image Text:n. Indicate ,r) f, COS TI) e coordinate 3. ons for the Q(-3, 5, 1) (u, v, w) phs 25. x = cos 8t, y sin 81, z = 26. x = cos't, y = sin²t, z = 1 0.8 81 120 27. Show that the curve with parametric equations x = 1 cos 1, y = f sin f, z = t lies on the cone z² = x² + y2, and use this fact to help sketch the curve. 28. Show that the curve with parametric equations x = sin 1, y = cost, z = sin²t is the curve of intersection of the surfaces z = x² and x² + y² = 1. Use this fact to help sketch the curve. 29. Find three different surfaces that contain the curve r(t) = 2ti + e'j + e¹¹ k. 21 30. Find three different surfaces that contain the curve r(t) = t²i+ In tj+ (1/1) k. 31. At what points does the curve r(t) = ti + (2t - 1²) k inter- sect the paraboloid z = x² + y²? 32. At what points does the helix r(t) = (sin t, cost, t) intersect the sphere x² + y² + z² = 5? 2 33-37 Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and view- points that reveal the true nature of the curve. 33. r(t) = (cost sin 2t, sin t sin 2t, cos 2t) 34. r(t) = (te', e, t) 35. r(t) = (sin 3t cos t, 4t, sin 3t sin t) 36. r(t) = (cos(8 cos t) sin t, sin(8 cos t) sin t, cost) 37. r(t) = (cos 2t, cos 3t, cos 4t) 38. Graph the curve with parametric equations x = sin f, y = sin 21, z = cos 4t. Explain its shape by graphing its projections onto the three coordinate planes. 39. Graph the curve with parametric equations x = (1 + cos 16t) cost
Expert Solution
Step 1

ans 31.

To determine the points of intersection between the curve r(t) = t i + (2t - t^2) k and the paraboloid z = x^2 + y^2, we need to find the values of t that satisfy both equations.

First, we can substitute the equation for the curve into the equation for the paraboloid to get an equation in terms of t:

z = x^2 + y^2 = (ti)^2 + [(2t - t^2)k]^2 = t^2 + 4t^2 - 4t^3 + t^4

Simplifying this equation, we get:

z = t^4 - 4t^3 + 5t^2

Now we can solve for the values of t that satisfy this equation. We can do this by setting z equal to the expression we just derived and solving for t:

t^4 - 4t^3 + 5t^2 - z = 0

This is a quartic equation, which can be difficult to solve. However, we can make use of the fact that the curve r(t) is a parametric equation of a line in 3D space, so we know that it intersects the paraboloid in at most two points. This means that the quartic equation we just derived has at most two real roots, and we can use this fact to simplify our calculations.

To find the values of t that satisfy the equation, we can use numerical methods such as Newton's method or the bisection method. Alternatively, we can graph the two equations in a 3D coordinate system and visually determine the points of intersection.

In summary, to find the points of intersection between the curve r(t) = t i + (2t - t^2) k and the paraboloid z = x^2 + y^2, we need to substitute the equation for the curve into the equation for the paraboloid to get an equation in terms of t, and then solve for the values of t that satisfy the equation. This can be done using numerical methods or by graphing the two equations in a 3D coordinate system.

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