Surfaces & Contour Maps - Calculus 3

1. A surface has the equation $z = f(x,y) = \sqrt{9 - x^2 - y^2}$ for $x^2 + y^2 ≤ 9$. When creating a contour map with level curves at $z = 0, 1, 2, 3$, which statement correctly describes the relationship between the areas enclosed by consecutive contour curves?
   Answer: The areas are $9\pi, 8\pi, 5\pi, 0$ because $r^2 = 9 - z^2$ for each level curve Setting $z = k$ gives $k = \sqrt{9 - x^2 - y^2}$, so $k^2 = 9 - x^2 - y^2$, or $x^2 + y^2 = 9 - k^2$. For $z = 0$: $r^2 = 9$, area = $9\pi$. For $z = 1$: $r^2 = 8$, area = $8\pi$. For $z = 2$: $r^2 = 5$, area = $5\pi$. For $z = 3$: $r^2 = 0$, area = $0$. Choice A is wrong because the radii differ. Choice B incorrectly states the decrease is linear in $z$ rather than in $z^2$. Choice D is wrong because this is a hemisphere, not a paraboloid.
2. Consider the surface $z = \sin(x) \cos(y)$. Examining the contour map in the region $0 ≤ x ≤ 2\pi, 0 ≤ y ≤ 2\pi$, what is the maximum number of times a single level curve $z = c$ (where $-1 < c < 1$, $c ≠ 0$) can intersect the line $y = \pi$?
   Answer: 2 times, occurring at the points where $\sin(x) = -c$ On the line $y = \pi$, we have $\cos(\pi) = -1$, so $z = \sin(x)(-1) = -\sin(x)$. For the level curve $z = c$, we need $-\sin(x) = c$, or $\sin(x) = -c$. In the interval $[0, 2\pi]$, the equation $\sin(x) = -c$ (where $-1 < c < 1$, $c ≠ 0$) has exactly 2 solutions. Choice B incorrectly considers both $\sin(x) = c$ and $\sin(x) = -c$. Choice C incorrectly concludes no intersections. Choice D overestimates the number of solutions.
3. A surface is defined by $z = x^2 - y^2 + 4$. If the contour map shows level curves at heights $z = 3, 4, 5, 6$, which statement best describes the behavior of these contours near the origin?
   Answer: The contours form hyperbolas that open along the $x$-axis for $z > 4$ and along the $y$-axis for $z < 4$ Setting $z = k$ gives $k = x^2 - y^2 + 4$, or $x^2 - y^2 = k - 4$. For $k = 3$: $x^2 - y^2 = -1$ (hyperbola opening along $y$-axis). For $k = 4$: $x^2 - y^2 = 0$ (intersecting lines $y = ±x$). For $k = 5, 6$: $x^2 - y^2 = 1, 2$ (hyperbolas opening along $x$-axis). Choice A is wrong because these are hyperbolas, not circles. Choice C is wrong because hyperbolas aren't ellipses. Choice D reverses the orientation.
4. Two surfaces are defined: Surface A by $z = x^2 + y^2$ and Surface B by $z = 2x^2 + 2y^2$. When comparing their contour maps at the same level heights $z = 2, 4, 6, 8$, what is the relationship between corresponding contour curves?
   Answer: Surface B's contours are circles with radii exactly $\frac{1}{\sqrt{2}}$ times those of Surface A's contours at each level For Surface A: $z = x^2 + y^2$ gives radius $r_A = \sqrt{z}$. For Surface B: $z = 2x^2 + 2y^2$ gives $x^2 + y^2 = z/2$, so radius $r_B = \sqrt{z/2} = \frac{\sqrt{z}}{\sqrt{2}} = \frac{r_A}{\sqrt{2}}$. Choice B reverses the relationship. Choice C is wrong because both surfaces give circular contours. Choice D misunderstands the scaling effect.

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