Convex pentagon PQRST has vertices P(0, 0), Q(-2, 2), R(-1, 4), S(x, y), and T(2, 2). QR = ST and RS = 2. Find (x, y).

Find the maximum and minimum values of $f(x, y)=y-x$ defined for the polygonal convex set having vertices at $(0,0),(4,0),(3,5),$ and $(0,5)$.

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.

Find the appropriate values of

$$n _ { 1 } \text { and } n _ { 2 }$$

(assume that

$$n _ { 1 } = n _ { 2 }$$

) needed to estimate

$$\left( \mu _ { 1 } - \mu _ { 2 } \right)$$

with A 90% confidence interval of width 1.0. Assume that

$$\sigma _ { 1 } ^ { 2 } \approx 5.8 \text { and } \sigma _ { 2 } ^ { 2 } \approx 7.5$$

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?

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 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.

Mark the statement True or False (T/F). Justify each answer.

If y = C1 V1 + C2 V2 + C3 V3 and Ct + C2 + C3 = 1, then y is a convex combination of v1, v2, and v3.

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.

$$x=-4 y^2+6 y+2$$

Write each equation in standard form. Identify the vertex, axis of symmetry, and direction of opening of the parabola.

$$y=2x^2-24x+40$$

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).

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.

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

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?