Prolonged exposure to mercury even at relatively low but toxic concentrations in the air is known to cause permanent mental disorders, insomnia, and pain and numbness in the hands and the feet, among other things. Therefore, the maximum allowable concentration of mercury vapor in the air at workplaces is regulated by federal agencies. These regulations require that the average level of mercury concentration in the air does not exceed $0.1 \mathrm{mg} / \mathrm{m}^{3}$. Consider a mercury spill that occurs in an airtight storage room at $20^{\circ} \mathrm{C}$ in San Francisco during an earthquake. Calculate the highest level of mercury concentration in the air that can occur in the storage room, in $\mathrm{mg} / \mathrm{m}^{3},$ and determine if it is within the safe level. The vapor pressure of mercury at $20^{\circ} \mathrm{C}$ is 0.173 Pa. Propose some guidelines to safeguard against the formation of toxic concentrations of mercury vapor in air in storage rooms and laboratories.

Evaluate the expression. Name the property used in each step.

$$7 + (9 - 3^2)$$

Use Lagrange multipliers to find the indicated extrema of f subject to two constraints. In each case, assume that x, y, and z are nonnegative. Maximize f(x, y, z) = xyz Constraints: x + y + z = 32, x - y + z = 0

Cornet Cable charges $32.50 a month for basic cable television. Each premium channel selected costs an additional$4.95 per month. Write an expression to find the cost of a month of cable service.

What compound is formed when limonene reacts with excess HBr? Is the compound chiral or achiral?

Solve the following initial value problem for u as a function of t:

$$\frac { d u } { d t } + \frac { k } { m } u = 0 \quad ( k \text { and } m \text { positive constants } ) , \quad u ( 0 ) = u _ { 0 }$$

a. as a first-order linear equation. b. as a separable equation.

(a) Determine the matrix inverse and condition number for the following matrix:

$$\left[\begin{array}{lll}{1} & {2} & {3} \\ {4} & {5} & {6} \\ {7} & {8} & {9}\end{array}\right]$$

(b) Repeat (a) but change $a_{33}$ slightly to 9.1.

Suppose that u and v are linearly independent vectors in $R^3$. Find the reduced row echelon form of A=$[a_1$ $a_2$ $a_3$ $a_4]$, given that

$$a_1=u$$

$$a_2=2u$$

$$a_3=u+v$$

$$a_4=v$$

A spaceship, moving away from Earth at a speed of 0.900c, reports back by transmitting at a frequency (measured in the spaceship frame) of 100 MHz. To what frequency must Earth receivers be tuned to receive the report?

Sketch the graph of a function f for which f(0)=-1, f'(0)=0, f'(x)<0 if x<0, and f'(x)>0 if x>0.

In R³ the orthogonal projections onto the x -axis, y-axis, and z-axis are

$$T_1(x, y, z) = (x, 0, 0), T_2(x, y, z) = (0, y, 0), T_3(x, y, z) = (0, 0, z)$$

Make a sketch showing x and x - T(x) in the case where T is the orthogonal projection on to the x -axis.

Evaluate the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ for the following vector fields F and curves C in two ways.

By parameterizing C .

Solve the equation.

$$( 34 x + 26 ) ^ { 1 / 3 } = 4 ( x - 1 ) ^ { 1 / 3 }$$

Use the Divergence Theorem to find the flux of F across the surface

$$\sigma$$

with outward orientation.

$$\mathbf { F } ( x , y , z ) = ( x - z ) \mathbf { i } + ( y - x ) \mathbf { j } + ( z - y ) \mathbf { k } ; \sigma$$

is the surface of the cylindrical solid bounded by

$$x ^ { 2 } + y ^ { 2 } = a ^ { 2 }$$

, z=0, and z=1.