Match each equation with its name. $$ \begin{array}{ll}-\vee\left(\frac{x}{6}\right)^{2}+\left(\frac{y}{3}\right)^{2}+\left(\frac{z}{7}\right)^{2}=1 & \text { a, ellipsoid } \ -\vee\left(\frac{x}{6}\right)^{2}-\left(\frac{y}{3}\right)^{2}-\left(\frac{z}{7}\right)^{2}=1 & \text { b. hyperboloid of two sheets } \ -\vee-\frac{x}{6}+\left(\frac{y}{3}\right)^{2}+\left(\frac{z}{7}\right)^{2}=0 & \text { c. elliptic paraboloid on } \mathrm{x} \text { axis } \ -\vee\left(\frac{x}{6}\right)^{2}-\left(\frac{y}{3}\right)^{2}+\left(\frac{z}{7}\right)^{2}=1 & \text { e. spherboloid of one sheet } \ -\vee\left(\frac{x}{6}\right)^{2}+\left(\frac{y}{6}\right)^{2}+\left(\frac{z}{6}\right)^{2}=1 & \end{array} $$

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To match each equation with its corresponding name, we need to analyze the structure of each equation. Here’s a breakdown of each equation:

1. **Equation:** $$-\vee\left(\frac{x}{6}\right)^{2}+\left(\frac{y}{3}\right)^{2}+\left(\frac{z}{7}\right)^{2}=1$$
   - This equation represents an **ellipsoid** because it has a positive term for $$y$$ and $$z$$ and a negative term for $$x$$, indicating a bounded surface.
   - **Match:** a. ellipsoid

2. **Equation:** $$-\vee\left(\frac{x}{6}\right)^{2}-\left(\frac{y}{3}\right)^{2}-\left(\frac{z}{7}\right)^{2}=1$$
   - This equation represents a **hyperboloid of two sheets** because it has all negative terms on the left side and equals a negative constant.
   - **Match:** b. hyperboloid of two sheets

3. **Equation:** $$-\vee-\frac{x}{6}+\left(\frac{y}{3}\right)^{2}+\left(\frac{z}{7}\right)^{2}=0$$
   - This equation represents an **elliptic paraboloid** because it has a quadratic form in $$y$$ and $$z$$ and a linear term in $$x$$.
   - **Match:** c. elliptic paraboloid on $$x$$ axis

4. **Equation:** $$-\vee\left(\frac{x}{6}\right)^{2}-\left(\frac{y}{3}\right)^{2}+\left(\frac{z}{7}\right)^{2}=1$$
   - This equation represents a **spherboloid of one sheet** because it has one positive term and two negative terms, indicating a surface that is unbounded in one direction.
   - **Match:** e. spherboloid of one sheet

5. **Equation:** $$-\vee\left(\frac{x}{6}\right)^{2}+\left(\frac{y}{6}\right)^{2}+\left(\frac{z}{6}\right)^{2}=1$$
   - This equation represents an **ellipsoid** as well, but it is not explicitly listed in the options. However, it can be inferred that it is another form of an ellipsoid.
   - **Match:** a. ellipsoid (this is a repeat)

Now, summarizing the matches:

- $$-\vee\left(\frac{x}{6}\right)^{2}+\left(\frac{y}{3}\right)^{2}+\left(\frac{z}{7}\right)^{2}=1$$ → a. ellipsoid
- $$-\vee\left(\frac{x}{6}\right)^{2}-\left(\frac{y}{3}\right)^{2}-\left(\frac{z}{7}\right)^{2}=1$$ → b. hyperboloid of two sheets
- $$-\vee-\frac{x}{6}+\left(\frac{y}{3}\right)^{2}+\left(\frac{z}{7}\right)^{2}=0$$ → c. elliptic paraboloid on $$x$$ axis
- $$-\vee\left(\frac{x}{6}\right)^{2}-\left(\frac{y}{3}\right)^{2}+\left(\frac{z}{7}\right)^{2}=1$$ → e. spherboloid of one sheet
- $$-\vee\left(\frac{x}{6}\right)^{2}+\left(\frac{y}{6}\right)^{2}+\left(\frac{z}{6}\right)^{2}=1$$ → a. ellipsoid (repeat)

Thus, the final matches are:

1. a
2. b
3. c
4. e
5. a (repeat)
1. a. ellipsoid 2. b. hyperboloid of two sheets 3. c. elliptic paraboloid on $$x$$ axis 4. e. spherboloid of one sheet 5. a. ellipsoid

Match each equation with its name.

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