Match the surfaces with the verbal description of the level curves by placing th
ID: 2894458 • Letter: M
Question
Match the surfaces with the verbal description of the level curves by placing the letter of the verbal description to the left of the number of the surface. 1. z = x^2 + y^2 2. z = xy 3. z = 2x + 3y 4. z = 2x^2 + 3y^2 5. z = 1/x - 1 6. z = Squareroot 25 - x^2 - y^2 7. z = Squareroot x^2 + y^2 A. two straight lines and a collection of hyperbolas B. a collection of concentric ellipses C. a collection of unequally spaced parallel lines D. a collection of unequally spaced concentric circles E. a collection of equally spaced parallel lines F. a collection of equally spaced concentric circlesExplanation / Answer
[1]
x^2 + y^2 = z is the equation for a circle of radius sqrt(z).They are unevenly spaced concentric circles, because the radii do not increase linearly with z. The answer is D
[2]
Rewriting, we have y = z/x. This is the equation of a hyperbola (except when z = 0). So the answer is A
[3]
Solving for y gives: y = (-2/3) x + z/3. This is the equation for a line of slope -2/3 with y-intercept z/3. These are parallel because they all have the same slope, and evenly-spaced because each time we increase z by 1 the y-intercept moves up by 1/3. The answer is E
[4]
(2/z) x^2 + (3/z) y^2 is an equation for an ellipse, so the answer is B
[5]
Solving for x, we have x = 1 + 1/z. These level curves are vertical lines, which are obviously parallel. However, they are unequally spaced, since the distance between them gets smaller as z gets larger. The answer is C
[6]
We can rewrite this as x^2 + y^2 = 25 - z^2. This is the equation for a circle, so the level curves are circles. However, they are unevenly spaced, because the radius of each circle is sqrt(25-z^2). So the answer is D
[7]
Rewriting, we have x^2 + y^2 = z^2. This is the equation for a circle of radius z, so the level curves are evenly spaced circles. The answer is F
Hope this Helps!!