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3 years ago

15

Factorise 1000a^2+27b^2

1 answer:

Ugo [173]3 years ago

5 0

1000a+27b with 2 on top of the number

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The intensity of light with wavelength λ traveling through a diffraction grating with N slits at an angle θ is given by I(θ) = N

Ymorist [56]

Answer:

0.007502795

Step-by-step explanation:

We have

N = 10,000

$\bf d=10^{-4}$

$\bf \lambda = 632.8*10^{-9}$

Replacing these values in the expression for k:

$\bf k=\frac{\pi Ndsin\theta}{\lambda}=\frac{\pi10^4*10^{-4}sin\theta}{632.8*10^{-9}}=\frac{\pi 10^9sin\theta}{632.8}$

So, the intensity is given by the function

$\bf I(\theta)=\frac{N^2sin^2(k)}{k^2}=\frac{10^8sin^2(\frac{\pi 10^9sin\theta}{632.8})}{(\frac{\pi 10^9sin\theta}{632.8})^2}$

The <em>total light intensity</em> is then

$\bf \int_{-10^{-6}}^{10^{-6}} I(\theta)d\theta=\int_{-10^{-6}}^{10{-6}}\frac{10^8sin^2(\frac{\pi 10^9sin\theta}{632.8})}{(\frac{\pi 10^9sin\theta}{632.8})^2}d\theta$

Since $\bf I(\theta)$ is an <em>even function</em>

$\bf \int_{-10^{-6}}^{10^{-6}} I(\theta)d\theta=2\int_{0}^{10^{-6}}I(\theta)d\theta$

and we only have to divide the interval $\bf [0,10^{-6}]$ in five equal sub-intervals $\bf I_1,I_2,I_3,I_4,I_5$ with midpoints $\bf m_1,m_2,m_3,m_4,m_5$

The sub-intervals and their midpoints are

$\bf I_1=[0,\frac{10^{-6}}{5}]\;,m_1=10^{-5}\\I_2=[\frac{10^{-6}}{5},2\frac{10^{-6}}{5}]\;,m_2=3*10^{-5}\\I_3=[2\frac{10^{-6}}{5},3\frac{10^{-6}}{5}]\;,m_3=5*10^{-5}\\I_4=[3\frac{10^{-6}}{5},4\frac{10^{-6}}{5}]\;,m_4=7*10^{-5}\\I_5=[4\frac{10^{-6}}{5},10^{-6}]\;,m_5=9*10^{-5}$

<em>By the midpoint rule</em>

$\bf \int_{0}^{10^{-6}}I(\theta)d\theta\approx\frac{10^{-6}}{5}[I(m_1)+I(m_2)+...+I(m_5)]$

computing the values of I:

$\bf I(m_1)=I(10^{-5})=\frac{10^8sin^2(\frac{\pi 10^9sin(10^{-5})}{632.8})}{(\frac{\pi 10^9sin(10^{-5})}{632.8})^2}=13681.31478$

$\bf I(m_2)=I(3*10^{-5})=\frac{10^8sin^2(\frac{\pi 10^9sin(3*10^{-5})}{632.8})}{(\frac{\pi 10^9sin(3*10^{-5})}{632.8})^2}=4144.509447$

Similarly with the help of a calculator or spreadsheet we find

$\bf I(m_3)=3.09562973\\I(m_4)=716.7480066\\I(m_5)=211.3187228$

and we have

$\bf \int_{0}^{10^{-6}}I(\theta)d\theta\approx\frac{10^{-6}}{5}[I(m_1)+I(m_2)+...+I(m_5)]=\frac{10^{-6}}{5}(18756.98654)=0.003751395$

Finally the the total light intensity

would be 2*0.003751395 = 0.007502795

8 0

3 years ago

Choose the equation that represents the line passing through the point (2, - 5) with a slope of −3.

zzz [600]

<h2>Answer:</h2>

We must determine the y-intercept, b using the slope and point provided.

$-5 = -3(2) + b\\\\-5 = -6 + b\\\\1 = b$

Now that we know the y-intercept, b, we can create the equation of this line: <em>y = -3x + 1</em>.

4 0

4 years ago

Find the value of each variable. If you give the decimal value, round to the nearest tenth.

DiKsa [7]

u = 10.4 and v = 12

Solution:

In the given 2 sides of a triangle are 60°, 60°.

Sum of all the angles of a triangle = 180°

60° + 60° + third angle = 180°

⇒ third angle = 180° – 60° – 60°

⇒ third angle = 60°

All angles are equal, therefore the given triangle is an equilateral triangle.

⇒ All sides are equal in length.

⇒ v = 12

The line drawn from the top angle divides the triangle into two equal parts

and the line is perpendicular.

12 ÷ 2 = 6

Using Pythagoras theorem,

$\text{Base}^2+\text{Opposite}^2=\text{Hypotenuse}^2$

⇒ $6^2+\text{opposite}^2=12^2$

⇒ $36+\text{opposite}^2=144$

⇒ $\text{opposite}^2=108$

⇒ $\text{opposite}=6\sqrt3=10.4$

⇒ u = 10.4

Hence, u = 10.4 and v = 12.

7 0

3 years ago

Y = 2x + 1<br> y = 4x - 1

STatiana [176]

Answer:

1

Step-by-step explanation:

Y = 2X +1

= 2(0)+1

=0 + 1

=1

7 0

4 years ago

The grid map shows the location of the houses of some of Kendra's friends. Which friend lives at

jenyasd209 [6]

Answer:

there is no picture and i think something is messed up because the choices dont make any sence

Step-by-step explanation:

6 0

3 years ago