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2 years ago

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Worksheet on the Law of sines

1 answer:

Amanda [17]2 years ago

3 0

PLEASE HELPPPPP ME PLEASE I DONT KNOW WHAT TO DO ☹️#8

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Solve y ' ' + 4 y = 0 , y ( 0 ) = 2 , y ' ( 0 ) = 2 The resulting oscillation will have Amplitude: Period: If your solution is A

Vlad [161]

Answer:

$y(x)=sin(2x)+2cos(2x)$

Step-by-step explanation:

$y''+4y=0$

This is a homogeneous linear equation. So, assume a solution will be proportional to:

$e^{\lambda x} \\\\for\hspace{3}some\hspace{3}constant\hspace{3}\lambda$

Now, substitute $y(x)=e^{\lambda x}$ into the differential equation:

$\frac{d^2}{dx^2} (e^{\lambda x} ) +4e^{\lambda x} =0$

Using the characteristic equation:

$\lambda ^2 e^{\lambda x} + 4e^{\lambda x} =0$

Factor out $e^{\lambda x}$

$e^{\lambda x}(\lambda ^2 +4) =0$

Where:

$e^{\lambda x} \neq 0\\\\for\hspace{3}any\hspace{3}\lambda$

Therefore the zeros must come from the polynomial:

$\lambda^2+4 =0$

Solving for $\lambda$ :

$\lambda =\pm2i$

These roots give the next solutions:

$y_1(x)=c_1 e^{2ix} \\\\and\\\\y_2(x)=c_2 e^{-2ix}$

Where $c_1$ and $c_2$ are arbitrary constants. Now, the general solution is the sum of the previous solutions:

$y(x)=c_1 e^{2ix} +c_2 e^{-2ix}$

Using Euler's identity:

$e^{\alpha +i\beta} =e^{\alpha} cos(\beta)+ie^{\alpha} sin(\beta)$

$y(x)=c_1 (cos(2x)+isin(2x))+c_2(cos(2x)-isin(2x))\\\\Regroup\\\\y(x)=(c_1+c_2)cos(2x) +i(c_1-c_2)sin(2x)\\$

Redefine:

$i(c_1-c_2)=c_1\\\\c_1+c_2=c_2$

Since these are arbitrary constants

$y(x)=c_1sin(2x)+c_2cos(2x)$

Now, let's find its derivative in order to find $c_1$ and $c_2$

$y'(x)=2c_1 cos(2x)-2c_2sin(2x)$

Evaluating $y(0)=2$ :

$y(0)=2=c_1sin(0)+c_2cos(0)\\\\2=c_2$

Evaluating $y'(0)=2$ :

$y'(0)=2=2c_1cos(0)-2c_2sin(0)\\\\2=2c_1\\\\c_1=1$

Finally, the solution is given by:

$y(x)=sin(2x)+2cos(2x)$

5 0

3 years ago

Which choice is the explicit formula for the following geometric sequence? 0.2, -0.06, 0.018, -0.0054, 0.00162

Crazy boy [7]

Answer:

aₙ = 0.2(-0.3)⁽ⁿ⁻¹⁾

Step-by-step explanation:

The geometric sequence with the first term a, a common ratio r has the nth term given as

Tₙ = arⁿ⁻¹

where Tₙ is the nth term

From the given sequence

a = 0.2

r = -0.06/0.2

= -0.3

Hence the nth term

= 0.2 * -0.3ⁿ⁻¹

The right option is E

8 0

2 years ago

You roll a die, winning nothing if the number of spots is odd, $5 for a 2 or a 4, and $20 for a 6. a) Find the expected va

Dovator [93]

TeX rendering has been iffy at best on this site for the past few days, at least in my experience. I've attached a solution below.

8 0

2 years ago

What is 3(x+2)=15?????

Nitella [24]

Answer:

3

Step-by-step explanation:

7 0

2 years ago

The length of the diagonal of a rectangle is √181 inches. Which statement describes the length of the diagonal?

lakkis [162]

Answer:

The correct option is (B).

Step-by-step explanation:

The length of the diagonal of a rectangle is $\sqrt{181}$ inches.

Compute the value of $\sqrt{181}$ inches as follows:

The number 181 is not a square of a whole number.

So, the square root of 181 must lie between two whole numbers.

Consider the following squares:

$12^{2}=144$

$13^{2}=169$

$14^{2}=196$

It is quite clear that the 181 lies between the square of 13 and 14.

So, it can be said that the square root of 181 is between the square of 13 and 14.

Thus, the length of the diagonal of a rectangle is between 13 and 14 inches.

The correct option is (B).

4 0

2 years ago