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=2(x^2)+3(y^2)2. z=√(25-x^2-y^2)3. z=xy4. z=√(x^2+y^2)5. z=1/(x-1)6. z=x^2+y^27. z=2x+3yA. A collection of concentric ellipsesB. Two straight lines and a collection ofhyperbolasC. A collection of equally spaced concentriccirclesD. A collection of equally spaced parallellinesE. A collection of unequally spaced parallellinesF. A collection of unequally spacedconcentric circles

Answer 1

Answer:

Step-by-step explanation:

1. (2/z)x^2 + (3/z)y^2 is an equation for an ellipse, so the answeris A.

2. We can rewrite this as x^2 + y^2 = 25 - z^2. This is theequation for a circle, so the level curves are circles. However,they are unevenly spaced, because the radius of each circle issqrt(25-z^2). So the answer is F.

3. Rewriting, we have y = z/x. This is the equation of a hyperbola(except when z = 0). So the answer is B.

4. Rewriting, we have x^2 + y^2 = z^2. This is the equation for acircle of radius z, so the level curves are evenly spaced circles.The answer is C.

5. Solving for x, we have x = 1 + 1/z. These level curves arevertical lines, which are obviously parallel. However, they areunequally spaced, since the distance between them gets smaller as zgets larger. The answer is E.

6. x^2 + y^2 = z is the equation for a circle of radius sqrt(z).They are unevenly spaced once again, because the radii do notincrease linearly with z. The answer is F.

7. Solving for y gives: y = (-2/3)x + z/3. This is the equation fora line of slope -2/3 with y-intercept z/3. These are parallelbecause they all have the same slope, and evenly-spaced becauseeach time we increase z by 1 the y-intercept moves up by 1/3. Theanswer is D.