In a paint mixing plant, two tanks supply fluids to a mixing cistern. The height, h, of the fluid in the cistern is dependent upon the difference between the input mass flow rate, q, and the output flow rate, $q_{e}$. A nonlinear differential equation describing this dependency is given by

$$\frac{d h}{d t}+\frac{A_{e}}{A} \sqrt{2 g h}=\frac{q}{\rho A}$$

where A = cross-sectional area of the cistern, $A_{c}$ = cross-sectional area of the exit pipe, g = acceleration due to gravity, and $\rho$ = liquid density. a. Linearize the nonlinear equation about the equilibrium point $\left(h_{0}, q_{0}\right)$ and find the transfer function relating the output cistern fluid level, H(s), to the input mass flow rate, Q(s). b. The color of the liquid in the cistern can be kept constant by adjusting the input flow rate, q, assuming the input flow's color is specifically controlled. Assuming an average height, $h_{a v},$ of the liquid in the cistern, the following equation relates the net flow of color to the cistern to the color in the cistern.

$$e_{1} q-e q_{e}=\frac{d}{d t}\left(\rho A e h_{a v}\right)$$

where $e_{1}$ = fractional part of flow representing color into the cistern, and e = fractional part of the cistern representing color in the cistern. Assume that the flow out of the cistern is constant and use the relationship,

$$q_{e}=\rho A_{e} \sqrt{2 g h_{a v}},$$

along with the given equation above to find the transfer function, E(s)/Q(s), that relates the color in the cistern to the input flow rate.

An electron in a TV CRT moves with a speed of $6.00 \times 10^{7} \mathrm{m} / \mathrm{s},$ in a direction perpendicular to the Earth’s field, which has a strength of $5.00 \times 10^{-5} \mathrm{T}.$ (a) What strength electric field must be applied perpendicular to the Earth’s field to make the electron moves in a straight line? (b) If this is done between plates separated by 1.00 cm, what is the voltage applied? (Note that TVs are usually surrounded by a ferromagnetic material to shield against external magnetic fields and avoid the need for such a correction.)

For the following exercise, assume that f(x) and g(x) are both differentiable functions with values as given in the following table. Use the following table to calculate the following derivatives.

$$\begin{matrix} \text{x} & \text{1} & \text{2} & \text{3} & \text{4}\\ \text{f(x)} & \text{3} & \text{5} & \text{−2} & \text{0}\\ \text{g(x)} & \text{2} & \text{3} & \text{−4} & \text{6}\\ \text{f'(x)} & \text{−1} & \text{7} & \text{8} & \text{−3}\\ \text{g'(x)} & \text{4} & \text{1} & \text{2} & \text{9}\\ \end{matrix}$$

Find h'(4) if $h(x)=\frac{1}{x}+\frac{g(x)}{f(x)}$

The current in a circuit element is $i(t)=4\left(1-e^{-5 t}\right)$ A when $t \geq 0$ and i(t)=0 when t<0. Determine the total charge that has entered a circuit element for $t \geq 0$. Hint: $q(0)=\int_{-\infty}^{0} i(\tau) d \tau=\int_{-\infty}^{0} 0 d \tau=0$

For the following exercise, find the definite or indefinite integral. $\int_{0}^{1} \frac{d x}{3+x}$

Why is a golfer with a metal club over her shoulder vulnerable to lightning in an open fairway? Would she be any safer under a tree?

Shao-Mei and Elisha, who are good friends, decide to donate blood. Shao-Mei discovers that her blood group is AB $Rh^−$ and Elisha’s blood group is O $Rh^−.$ Explain how Elisha could help Shao-Mei if she ever needed a blood transfusion. Why cannot Shao-Mei help Elisha in the same way?

The matrices and vectors listed in Eq. are used in several of the exercises that follow.

$$A=\left[\begin{array}{ll} 2 & 3 \\ 1 & 4 \end{array}\right], \ B=\left[\begin{array}{ll} 1 & 2 \\ 1 & 4 \end{array}\right], \ \mathbf{u}=\left[\begin{array}{l} 1 \\ 3 \end{array}\right]$$

,

$$\mathbf{v}=\left[\begin{array}{ll} 2,4 \end{array}\right], \ C=\left[\begin{array}{rr} 2 & 1 \\ 4 & 0 \\ 8 & -1 \\ 3 & 2 \end{array}\right]$$

,

$$D=\left[\begin{array}{rrrr} 2 & 1 & 3 & 6 \\ 2 & 0 & 0 & 4 \\ 1 & -1 & 1 & -1 \\ 1 & 3 & 1 & 2 \end{array}\right], \ \mathbf{w}=\left[\begin{array}{r} 2 \\ 3 \\ 1 \\ 1 \end{array}\right]$$

Find each of the following. uv and vu

A matrix A is given. a. Find the characteristic equation for A. b. Find the eigenvalues of A. c. Find the eigenvectors corresponding to each eigenvalue. d. Verify the result of part (c) by showing that $A \mathbf{v}_{i}=\lambda_{i} \mathbf{v}_{i}$.

$$A=\left[\begin{array}{lll} 1 & 1 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]$$

Use the fact that nonempty open sets of an affine variety are everywhere denseto prove that an affine variety is connected in the Zariski topology. (A topological space is connected if it is not the union of two disjoint, proper, open subsets.)

How does hybridization of the substituted carbon atom change when an alcohol is converted into an aldehyde? An aldehyde to a carboxylic acid?

Which base, $\mathrm{CH}_{3} \mathrm{NH}_{2}$ or $\left(\mathrm{CH}_{3}\right)_{2} \mathrm{NH}$, is the stronger base? Which conjugate acid, $\left(\mathrm{CH}_{3}\right)_{2} \mathrm{NH}_{2}^+$ or $\left(\mathrm{CH}_{3}\right)_{2} \mathrm{NH}_{3}^+$, is the stronger acid?

(a) How long will the energy in a 1470-kJ (350-kcal) cup of yogurt last in a woman doing work at the rate of 150 W with an efficiency of 20.0% (such as in leisurely climbing stairs)? (b) Does the time found in part (a) imply that it is easy to consume more food energy than you can reasonably expect to work off with exercise?

What is the conjugate acid of each of the following? What is the conjugate base of each? (a) $\mathrm{OH}^{-}$ (b) $\mathrm{H}_{2} \mathrm{O}$ (c) $\mathrm{HCO}_{3}^{-}$ (d) $\mathrm{NH}_{3}$ (e) $\mathrm{HSO}_{4}^{-}$ (f) $\mathrm{H}_{2} \mathrm{O}_{2}$ (g) $\mathrm{HS}^{-}$ (h) $\mathrm{H}_{5} \mathrm{N}_{2}^{+}$