Can you explain three different ways we can write multiplication with a<br> variable?

What is the probability of tossing two heads

vodka [1.7K]

Hello!

Each time, there will be a 1/2 chance of tossing a head. Therefore, we multiply our two probabilities.

1/2(1/2)=1/4

Therefore, we have a 1/4 or 25% chance of tossing two heads in a row.

I hope this helps!

3 0

4 years ago

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I need to get both into y=mx+b

quester [9]

Isolate the y for both questions. Note that what you do to one side, you do to the other. Do the opposite of PEMDAS.

First question: x - 5y ≥ 3

First, subtract x from both sides

x (-x) - 5y ≥ 3 (-x)

-5y ≥ -x + 3

Divide -5 from both sides. Note that when you divide by a negative number, you flip the equation.

(-5y)/-5 ≥ (-x + 3)/-5

y ≤ (-x + 3)/-5

y ≤ x/5 - 0.6

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Do the same for the second one. Subtract x from both sides

x + y < 0

x (-x) + y < 0 (-x)

y < - x

~

5 0

3 years ago

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What are the intercepts of the equation <br> 2x+3/2y+3z=6

Mrrafil [7]

X-intercept => 2x = 6 => x = 3
x-intercept is 3.

y-intercept => 3/2y = 6 => y = 4
y-intercept is 4.

z-intercept => 3z = 6 => z = 2
z-intercept is 2.

3 0

3 years ago

50=5x−3−8x−7<br> what is X?

Black_prince [1.1K]

Answer:

x = -20

Step-by-step explanation:

Solve for x:

50 = 5 x - 8 x - 7 - 3

Grouping like terms, 5 x - 8 x - 7 - 3 = (5 x - 8 x) + (-3 - 7):

50 = (5 x - 8 x) + (-3 - 7)

5 x - 8 x = -3 x:

50 = -3 x + (-3 - 7)

-3 - 7 = -10:

50 = -10 - 3 x

50 = -3 x - 10 is equivalent to -3 x - 10 = 50:

-3 x - 10 = 50

Add 10 to both sides:

(10 - 10) - 3 x = 10 + 50

10 - 10 = 0:

-3 x = 50 + 10

50 + 10 = 60:

-3 x = 60

Divide both sides of -3 x = 60 by -3:

(-3 x)/(-3) = 60/(-3)

(-3)/(-3) = 1:

x = 60/(-3)

The gcd of 60 and -3 is 3, so 60/(-3) = (3×20)/(3 (-1)) = 3/3×20/(-1) = 20/(-1):

x = 20/(-1)

Multiply numerator and denominator of 20/(-1) by -1:

Answer: x = -20

4 0

3 years ago

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Let C be the curve of intersection of the parabolic cylinder x^2 = 2y, and the surface 3z = xy. Find the exact length of C from

Maslowich

I've attached a plot of the intersection (highlighted in red) between the parabolic cylinder (orange) and the hyperbolic paraboloid (blue).

The arc length can be computed with a line integral, but first we'll need a parameterization for $C$ . This is easy enough to do. First fix any one variable. For convenience, choose $x$ .

Now, $x^2=2y\implies y=\dfrac{x^2}2$ , and $3z=xy\implies z=\dfrac{x^3}6$ . The intersection is thus parameterized by the vector-valued function

$\mathbf r(x)=\left\langle x,\dfrac{x^2}2,\dfrac{x^3}6\right\rangle$

where $0\le x\le 4$ . The arc length is computed with the integral

$\displaystyle\int_C\mathrm dS=\int_0^4\|\mathbf r'(x)\|\,\mathrm dx=\int_0^4\sqrt{x^2+\dfrac{x^4}4+\dfrac{x^6}{36}}\,\mathrm dx$

Some rewriting:

$\sqrt{x^2+\dfrac{x^4}4+\dfrac{x^6}{36}}=\sqrt{\dfrac{x^2}{36}}\sqrt{x^4+9x^2+36}=\dfrac x6\sqrt{x^4+9x^2+36}$

Complete the square to get

$x^4+9x^2+36=\left(x^2+\dfrac92\right)^2+\dfrac{63}4$

So in the integral, you can substitute $y=x^2+\dfrac92$ to get

$\displaystyle\frac16\int_0^4x\sqrt{\left(x^2+\frac92\right)^2+\frac{63}4}\,\mathrm dx=\frac1{12}\int_{9/2}^{41/2}\sqrt{y^2+\frac{63}4}\,\mathrm dy$

Next substitute $y=\dfrac{\sqrt{63}}2\tan z$ , so that the integral becomes

$\displaystyle\frac1{12}\int_{9/2}^{41/2}\sqrt{y^2+\frac{63}4}\,\mathrm dy=\frac{21}{16}\int_{\arctan(3/\sqrt7)}^{\arctan(41/(3\sqrt7))}\sec^3z\,\mathrm dz$

This is a fairly standard integral (it even has its own Wiki page, if you're not familiar with the derivation):

$\displaystyle\int\sec^3z\,\mathrm dz=\frac12\sec z\tan z+\frac12\ln|\sec x+\tan x|+C$

So the arc length is

$\displaystyle\frac{21}{32}\left(\sec z\tan z+\ln|\sec x+\tan x|\right)\bigg|_{z=\arctan(3/\sqrt7)}^{z=\arctan(41/(3\sqrt7))}=\frac{21}{32}\ln\left(\frac{41+4\sqrt{109}}{21}\right)+\frac{41\sqrt{109}}{24}-\frac98$

4 0

4 years ago

Can you explain three different ways we can write multiplication with a variable?