Solve the given differential equation by separation of variables.

$$\csc y d x + \sec ^ { 2 } x d y = 0$$

Simplify the expression. Assume that all variables are positive.

$$\sqrt [ 3 ] { \frac { 125 x ^ { 6 } } { 6 } }$$

(III) On a level billiards table a cue ball, initially at rest at point $\mathrm{O}$ on the table, is struck so that it leaves the cue stick with a center-of-mass speed $v_0$ and a "reverse" spin of angular speed $\omega_0$ (see Fig. 11-41). A kinetic friction force acts on the ball as it initially skids across the table. If $\omega_0$ is $10 \%$ smaller than $\omega_{\mathrm{C}}$, i.e., $\omega_0=0.90 \omega_{\mathrm{C}}$, determine the ball's $\mathrm{CM}$ velocity $v_{\mathrm{CM}}$ when it starts to roll without slipping.

Simplify the expression. State the excluded values of the variables.

$$\frac { z ^ { 2 } + z - 2 } { z ^ { 2 } - 3 z + 2 }$$

Un voyage. Qu'est-ce que ces personnes ont fait dans le sud de la France? Faites des phrases completes au passé composé.

MODELE: nous / choisir / huitres

Nous avons choisi des huitres.

toi et moi, nous / boire / coca-cola / café

Write each expression in simplest form. Assume that all variables are positive.

$$8^(2/3)$$

Lee los carteles. Imagina que eres el cliente y después de ir de compras, contesta las preguntas. -Corbatas: $29,99, Descuento del 50% si compra tres o más. -¡Muy barato!$12,99, Si paga con tarjeta, $13,50. -¡Increíble!,$100, Solo números 42 a 48.

1.¿Cuántas corbatas compraste? ¿Cuánto pagaste por ellas?. 2.¿Compraste algún collar? ¿Para quién? ¿Pagaste con tarjeta o en efectivo?. 3.¿Conseguiste zapatos de tu número? ¿Cuánto te costaron?___.

A matrix is given. (a) Determine whether the matrix is in row-echelon form. (b) Determine whether the matrix is in reduced row-echelon form. (c) Write the system of equations for which the given matrix is the augmented matrix.

$$\left[\begin{array}{rrr}1 & 0 & -3 \\ 0 & 1 & 5\end{array}\right]$$

A fire ant, searching for hot sauce in a picnic area, goes through three displacements along level ground: vector d1 for 0.40 m southwest (that is, at 45 degree from directly south and from directly west), vector d2 for 0.50 m due east, vector d3 for 0.60 m at 60 degree north of east. Let the positive x direction be east and the positive y direction be north.

If the ant is to return directly to the starting point how far should it move?

In 2004, Georgia had 6 income tax brackets. The tax rate on every collar of income was:

• $1 \%$ for incomes more than $\$ 0$ and up to and including $\$ 750$.
• $2 \%$ for incomes more than $\$ 750$ and up to and including $\$ 2250$,
• $3 \%$ for incomes more than $\$ 2250$ and up to and including $\$ 3750$,
• $4 \%$ for incomes more than $\$ 3750$ and up to and including $\$ 5250$,
• $5 \%$ for incomes more than $\$ 5250$ and up to and including $\$ 7000$,
• $6 \%$ for incomes more than $\$ 7000$.

Analyze the function you wrote to represent the tax brackets in Georgia from Question 1.

a. Would the lines that represent each piece of the function on the graph be connected or contain breaks? Explain your reasoning.

b. Describe the endpoints of the line representing each interval. Explain your reasoning.

An object in rectilinear motion moves along a horizontal line with the positive direction to the right. The position s of the object from the origin at time $t \geq 0$ (in seconds) is given by the function s=s(t). (a) Determine the intervals during which the object moves to the right and the intervals during which it moves to the left. (b) When does the object reverse direction? (c) When is the velocity of the object increasing and when is it decreasing? (d) Draw a figure to illustrate the motion of the object. (e) Draw a figure to illustrate the velocity of the object. $s=2 t-\frac{6}{t}, \quad t>0$

If the collar $P$ is given a constant acceleration $a=$ $3 g$ to the right, the pendulum will assume a steady-state deflection $\theta=30^{\circ}$. Determine the stiffness $k_T$ of the torsional spring which will allow this to happen. The torsional spring is undeformed when the pendulum is in the vertical position.

In order to determine controllability mathematically, the controllability matrix is formed, and its rank evaluated. What is the final step in determining controllability if the controllability matrix is a square matrix?

What are allotropes?