Consider a bead that is threaded on a rigid circular hoop of radius R lying in the xy plane with its center at O, and use the angle φ of two-dimensional polar coordinates as the one generalized coordinate to describe the bead's position. Write down the equations that give the Cartesian coordinates (x, y) in terms of φ and the equation that gives the generalized coordinate φ in terms of (x, y).

What do we mean when we say that a compound is neutral?

Fix a positive integer n. The Laplacian $\Delta p$ of a twice differentiable function p on $\mathbf{R}^{n}$ is the function on $\mathbf{R}^{n}$ defined by $\Delta p=\frac{\partial^{2} p}{\partial x_{1}^{2}}+\cdots+\frac{\partial^{2} p}{\partial x_{n}^{2}}.$ The function p is called harmonic if $\Delta p=0.$ A polynomial on $\mathbf{R}^{n}$ is a linear combination of functions of the form $x_{1}^{m_{1}} \cdots x_{n}^{m_{n}},$ where $m_{1}, \dots, m_{n}$ are nonnegative integers. Suppose q is a polynomial on $\mathbf{R}^{n}.$ Prove that there exists a harmonic polynomial p on $\mathbf{R}^{n}$ such that $p(x)=q(x)$ for every $x \in \mathbf{R}^{n}$ with $\|x\|=1,$ then p = 0.]

Develop a function to produce an animation of a particle moving in a circle in Cartesian coordinates based on radial coordinates. Assume a constant radius, r, and allow the angle, $\theta$, to increase from zero to 2$\pi$ in equal increments. The function’s first lines should be function phasor(r, nt, nm) % function to show the orbit of a phasor % r = radius % nt = number of increments for theta % nm = number of movies Test your function with phasor(1, 256, 10)

A family uses 96 kilolitres of water in 90 days. a. Find the rate in litres per day. b. If the water board charges 65 cents per kilolitres, how much do they pay for the water used: i. over the 90 day period ii. per day?

Due to the increased time and distance of travel of a projectile, discuss potential adjustments you need to play each of the following sports on the Moon:

a) football

b) gymnastics

c) trapeze

d) baseball

For a binomial probability distribution, it is unusual for the number of successes to be less than $\mu-2.5 \sigma$ or greater than $\mu+2.5 \sigma$. For a binomial experiment with 10 trials for which the probability of success on a single trial is 0.2, is it unusual to have more than five successes? Explain.

Solve the double inequality:

$\frac{1}{2} x-3<5 x$ or $\frac{2}{5} x-5>6 x$

Find the exact value of the expression by using the double-angle or half-angle formulas.

$$\sin 165^{\circ}$$

Solve the compound inequality.

$$-2<\frac{1}{2} x-5<1$$

Given sentence below refers to a numbered sentence in the passage. Write the letter of the choice that gives the sentence a meaning that is closest to the original sentence. ________ By today's standards, the authorities involved in his trial would be acccused of __________ .
a. unfair delays
b. long speeches
c. entertainment
d. official misconduct

A particle performs simple harmonic oscillations with amplitude 0.1 mm and frequency 100 Hz. What is the maximum acceleration of this particle

$$\left( \text { in } \mathrm { ms } ^ { - 2 } \right)$$

? A.

$$0.2 \pi$$

B.

$$0.4 \pi$$

C.

$$2 \pi ^ { 2 }$$

D.

$$4 \pi ^ { 2 }$$

Use the change-of-base formula and a calculator to approximate each logarithm accurate to six significant digits.

$$\log _3 40$$

The Cartesian coordinates of a point are given. (-4, 4) (i) Find polar coordinates (r, $\theta$) of the point, where r > 0 and $0 \leqslant \theta<2 \pi$. (ii) Find polar coordinates (r, $\theta$) of the point, where r < 0 and $0 \leqslant \theta<2 \pi$.