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3 years ago

12

Phong is rolling a number cube and flipping a coin .what is the probability that the cube will land on 4 and the coin will land

heads up.

1 answer:

Grace [21]3 years ago

3 0

Probabblity that the cube lands on 4 = 1/6
Probablity that the coin lands heads up = 1/2
Probablity of both happening = 1/2 * 1/6 = 1/12

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Simplify the complex numbers using de Moivre's Theorem and match them with their solutions.

alexandr1967 [171]

For the numbers in $a+bi$ form, convert to polar form:

$1+i=\sqrt2\dfrac{1+i}{\sqrt2}=\sqrt2\left(\cos\dfrac\pi4+i\sin\dfrac\pi4\right)$

By DeMoivre's theorem,

$(1+i)^5=(\sqrt2)^5\left(\cos\dfrac{5\pi}4+i\sin\dfrac{5\pi}4\right)=4\sqrt2\dfrac{-1-i}{\sqrt2}=-4-4i$

$-1+i=\sqrt2\dfrac{-1+i}{\sqrt2}=\sqrt2\left(\cos\dfrac{3\pi}4+i\sin\dfrac{3\pi}4}\right)$

$\implies(-1+i)^6=(\sqrt2)^6\left(\cos\dfrac{18\pi}4+i\sin\dfrac{18\pi}4\right)=8i$

$\sqrt3+i=2\dfrac{\sqrt3+i}2=2\left(\cos\dfrac\pi6+i\sin\dfrac\pi6\right)$

$\implies2(\sqrt3+i)^{10}=2^{11}\left(\cos\dfrac{10\pi}6+i\sin\dfrac{10\pi}6\right)=2^{11}\dfrac{1-i\sqrt3}2=2^{10}(1-i\sqrt3)$

For the numbers already in polar form, DeMoivre's theorem can be applied directly:

$2\left(\cos20^\circ+i\sin20^\circ\right)^3=2\left(\cos60^\circ+i\sin60^\circ\right)=2\dfrac{1+i\sqrt3}2=1+i\sqrt3$

$2\left(\cos\dfrac\pi4+i\sin\dfrac\pi4\right)^4=2(\cos\pi+i\sin\pi)=-2$

At second glance, I think the 2s in the last two numbers should also be getting raised to the 3rd and 4th powers:

$\left(2(\cos20^\circ+i\sin20^\circ)\right)^3=8\left(\cos60^\circ+i\sin60^\circ\right)=4+4\sqrt3$

$\left(2\left(\cos\dfrac\pi4+i\sin\dfrac\pi4\right)\right)^4=16(\cos\pi+i\sin\pi)=-16$

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2 years ago

The perimeter of one of the rectangular playrooms at Miss Pittypat's Day Care is 72 feet. It's width is 10 feet.

Alex17521 [72]

Answer:

The area of the rectangular playroom is 260 feet² .

Step-by-step explanation:

Given:

The perimeter of the rectangular playroom is 72 feet and it's width is 10 feet.

Now, to find its area we need the length .

So, putting the formula of perimeter we get it's length:

Let the length be l .

Perimeter = 2( length + width)

$72=2(l+10)$

$72=2l+20$

subtracting both sides from 20:

$72-20=2l+20-20$

$52=2l$

Dividing both sides by 2:

$52\div2=2l\div2$

$26=l$

Length is 26 feet.

And, to get the area we put the formula of finding the area:

$Area=length\times width$

$Area=26\times 10$

$Area=260$

Therefore, the area of the rectangular playroom is 260 feet² .

3 0

3 years ago

Antonia bought a set of dishes, a towel, and a candle at a housewares store. The set of dishes cost $35.02, the towel cost $21.4

vekshin1

Answer:

Antonia saved $5.39

Step-by-step explanation:

Candle:

$9.59 - $0.50 (coupon) = $9.09

Towel:

$21.43 - $2.75 (coupon) = $18.68

Dishes:

$21.43 x 10% = 2.143

$21.43 - $2.143 = $19.287, rounded to the nearest cent and it’s $19.29.

Calculate the amount of money saved by adding 0.50 + 2.75 + 2.143, which is 5.393. Round to the nearest cent and it will be $5.39.

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2 years ago

(4x-5)+2(3x+1)<br> Can you help me solve this plz for me

Korvikt [17]

Answer:

10x - 3

Step-by-step explanation:

First, distribute 2 to all terms within the second parenthesis:

2(3x + 1) = (2 * 3x) + (2 * 1) = 6x + 2

4x - 5 + 6x + 2

Combine like terms (terms with the same amount of variables).

4x + 6x + 2 - 5

(4x + 6x) + (2 - 5)

10x + (-3)

10x - 3

10x - 3 is your answer.

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2 years ago

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Sunny_sXe [5.5K]

Answer: x=5

Step-by-step explanation:

Substitute 17 for n(x)

17=2x+7

Subtract 7 from both sides

10=2x

Divide each side by 2 to get the x by itself

x=5

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2 years ago