For a concert, a circular stage is constructed and placed in the center so fans can completely surround the musicians. To record the sound of the crowd, two directional microphones are placed next to each other on the stage, one facing due east and the other facing due west. The patterns of the microphones can be represented by the polar equations $r=2.5+2.5 \cos \theta$ and $r=-2.5-2.5 \cos \theta.$ Identify the type of curve given by each polar equation.

Liquid sodium is being considered as an engine coolant. How many grams of liquid sodium are needed to absorb 1.00 MJ of energy in the form of heat if the temperature of the sodium is not to increase by more than $10^{\circ} \mathrm{C}$? Take $C_{\mathrm{P}}=30.8 \mathrm{J} \cdot \mathrm{K}^{-1} \cdot \mathrm{mol}^{-1}$ for Na(l ) at 500 K.

Use the properties of logarithms to rewrite each logarithm below in the form a ln 2 + b ln 3, where a and b are constants. Then approximate the value of each logarithm given that $\ln 2 \approx 0.69$ and $\ln 3 \approx 1.10.$ $\ln 24$

By referring to the periodic table, classify each of the following as a main-group element, a transition metal, or an inner transition metal. If a main-group or transition metal element, indicate which group and whether the element is a metal, a nonmetal, or a semimetal. $Se \quad As \quad Mo \quad Rn \quad Ta \quad \mathrm{Bi} \quad \mathrm{In}$

Let Y be an exponential random variable, where

$$f_{Y}(y)=\lambda e^{-\lambda y}, 0 \leq y.$$

For any positive integer n, show that

$$P(n \leq Y \leq n+1)=e^{-\lambda n}\left(1-e^{-\lambda}\right) .$$

Note that if $p=1-e^{-\lambda},$ the "discrete" version of the exponential pdf is the geometric pdf.

Describe the possible geometric isomers of $\mathrm{C}_{2} \mathrm{H}_{4} \mathrm{Cl}_{2}$.

Calculate the pressures in atmospheres at which the mean free path of a hydrogen molecule will be 1.00 $\mu \mathrm{m}$, 1.00 mm, and 1.00 m at $20.0^{\circ} \mathrm{C}$.

Calculate the number of moles in (a) 52.0 grams of calcium carbonate

Write each equation in the $x^{\prime} y^{\prime}$-plane for the given value of $\theta.$ Then identify the conic. $x^{2}-2 x+y=5 ; \theta=\frac{\pi}{3}$

Suppose a light has a brightness intensity of $I_1$ when it is at a distance of $d_1$ and a brightness intensity of $I_2$ when it is at a distance of $d_2$. These quantities are related by the equation $\frac{d_2}{d_1}=\sqrt{\frac{l_1}{l_2}}$. Suppose $I_1=50$ units and $I_2=24$ units. What would $\frac{d_2}{d_1}$ be? Express your answer in simplest form.

Solve each equation. $2 \log _{5}(x-2)=\log _{5} 36$

A mounted surveillance camera oscillates and views part of a circular region determined by $-60^{\circ} \leq \theta \leq 150^{\circ}$ and $0 \leq r \leq 40$ where r is in meters. Sketch a graph of the region that the security camera can view on a polar grid.

The input signal to an audio amplifier is described by $v_{I}=(0.5+0.25 \sin 1200 \pi t)$ V, and the output is described by $v_{O}=(2+4 \sin 1200 \pi t+0.8 \sin 2400 \pi t+0.4 \sin 3600 \pi t)$ V. What is the voltage gain of the amplifier? What order harmonics are present in the signal? What is the total harmonic distortion in the output signal?

Arguments have been advanced for trade restrictions. List and explain the arguments in favor of trade restrictions. Then explain the criticism of each argument. Argument: ______, Explanation: ________ Criticium: ___________.