Find an equation of the plane tangent to the sphere $(x-1)^{2}+(y+2)^{2}+(z-2)^{2}=6$ at the point (2,-1,0).

A boy is standing near a convex mirror in a fun house at a fair. He notices that his image appears to be 0.60 m tall. If the magnification of the mirror is $\frac{1}{3}$, what is the boy’s height?

Use the following information to find the surface area of the specified objects. Round to the nearest square unit. A sphere is a solid formed by all points in space that are the same distance from a fixed point (the center). The formula for the surface area of a sphere is S = 4πr², where r is the radius. The radius of a spherical soap bubble is 0.5 inch.

Write the quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening.

$$y=\frac{1}{3}(x-1)^2+2$$

Use Euler's Formula for Polyhedra to answer the following questions.

a. If a convex polyhedron has 10 vertices and 8 faces, how many edges does it have?

b. Can a convex polyhedron that has 8 faces, 12 edges, and 7 vertices be constructed? Give reasons for your answer.

The graph above shows the number of suburban dwellers in the United States as a percentage of the total population. Study the data displayed in the graph and answer the questions below.
a.What trend in the percentage of suburban dwellers does this graph show?
b.How might the trend of suburban dwellers shown on this graph have affected life in suburbs and cities?

Mean Value Theorem for Several Variables If f is differentiable at each point of the line segment from a to b, then there exists on that line segment a point $\mathbf{c}$ between $\mathbf{a}$ and $\mathbf{b}$ such that

$$f(\mathbf{b})-f(\mathbf{a})=\nabla f(\mathbf{c}) \cdot(\mathbf{b}-\mathbf{a})$$

Assuming that this result is true, show that, if f is differentiable on a convex set S and if $\nabla f(\mathbf{p})=\mathbf{0}$ on S, then f is constant on S. Note: A set S is convex if each pair of points in S can be connected by a line segment in S.

Determine the size and location of an image produced by a convex lens with a focal length of $14.5 \mathrm{~cm}$ of an object $10.5 \mathrm{~cm}$ from the lens and $2.35 \mathrm{~cm}$ high?

Where would 129 be on a mod 13 clock (which goes from 0 to 12 )? What is the remainder when 129 is divided by 13?

Write each series using sigma notation. Find the sum. - 7 - 42 - 252 - ... - 54.432

Water enters a 5-ft-wide, 1-ft-deep channel as shown in the figure. Across the inlet the water velocity is $6 \mathrm{ft} / \mathrm{s}$ in the center portion of the channel and $1 \mathrm{ft} / \mathrm{s}$ in the remainder of it. Farther downstream the water flows at a uniform $2 \mathrm{ft} / \mathrm{s}$ velocity across the entire channel. The fixed control volume $A B C D$ coincides with the system at time $t=0$. Make a sketch to indicate (a) the system at time $t=0.5 \mathrm{~s}$ and (b) the fluid that has entered and exited the control volume in that time period.

Find the maximum and minimum values of the function $f(x, y)=3 x-y+4$ for the polygonal convex set determined by the system of inequalities.

$$\begin{array}{l} x+y \leq 8 \\ y \geq 3 \\ x \geq 2 \end{array}$$

Identify the coordinates of the vertex and focus, the equations of the axis of symmetry and directrix, and the direction of opening of the parabola with the given equation. Then find the length of the latus rectum and graph the parabola.

$$y=-2 x^2+5 x-10$$

Fill in the blanks.

$$\begin{matrix} \text{5th:} & \text{\_\_\_\_\_\_\_}\\ \text{3rd:} & \text{\_\_\_\_\_\_\_}\\ \text{Root:} & A\flat\\ \text{Type:} & \text{m}\\ \end{matrix}$$