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2 years ago

Determine the equation of the line that is perpendicular to the given line, through the given point.

Mathematics

1 answer:

exis [7]2 years ago

4 0

Answer:

The equation of the line is $y + 7 = -\frac{1}{3}(x - 9)$

Step-by-step explanation:

Equation of a line:

The equation of line, in point-slope form, is given by:

$y - y_0 = m(x - x_0)$

In which m is the slope and the point is $(x_0, y_0)$

Perpendicular lines:

If two lines are perpendicular, the multiplication of their slopes is -1.

Through (9,-7)

This means that $x_0 = 9, y_0 = -7$

$y - y_0 = m(x - x_0)$

$y - (-7) = m(x - 9)$

$y + 7 = m(x - 9)$

Perpendicular to y = 3x + 4

This line has slope 3, so:

$3m = -1$

$m = -\frac{1}{3}$

Thus

$y + 7 = m(x - 9)$

$y + 7 = -\frac{1}{3}(x - 9)$

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mr Goodwill [35]

From the table, y varies directly as x. Therefore, the constant of variation is k = 2 and y = 2x.

<h3>How to find a direct variation?</h3>

For direct variation,

y ∝ x

Therefore,

y = kx

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k = constant of proportionality

Hence,

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2 years ago

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3 years ago

Read 2 more answers

In triangle EFG, angle E equals 8x-16, angle F equals x+8, and angle G equals 2x-10. What is the value of x?

Stels [109]

Answer: The value of x= 18

Step-by-step explanation:

Given: In triangle EFG, angle E equals 8x-16, angle F equals x+8, and angle G equals 2x-10.

The sum of all angles of a triangle is 180°.

Therefore, in triangle EFG

∠E + ∠F +∠G= 180°

⇒ 8x-16 +x+8+ 2x-10 = 180

⇒ 11x-18= 180

⇒ 11x = 180+18

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Hence, the value of x= 18

6 0

2 years ago

Find the surface area of x^2+y^2+z^2=9 that lies above the cone z= sqrt(x^@+y^2)

Mashcka [7]

The cone equation gives

$z=\sqrt{x^2+y^2}\implies z^2=x^2+y^2$

which means that the intersection of the cone and sphere occurs at

$x^2+y^2+(x^2+y^2)=9\implies x^2+y^2=\dfrac92$

i.e. along the vertical cylinder of radius $\dfrac3{\sqrt2}$ when $z=\dfrac3{\sqrt2}$ .

We can parameterize the spherical cap in spherical coordinates by

$\mathbf r(\theta,\varphi)=\langle3\cos\theta\sin\varphi,3\sin\theta\sin\varphi,3\cos\varphi\right\rangle$

where $0\le\theta\le2\pi$ and $0\le\varphi\le\dfrac\pi4$ , which follows from the fact that the radius of the sphere is 3 and the height at which the sphere and cone intersect is $\dfrac3{\sqrt2}$ . So the angle between the vertical line through the origin and any line through the origin normal to the sphere along the cone's surface is

$\varphi=\cos^{-1}\left(\dfrac{\frac3{\sqrt2}}3\right)=\cos^{-1}\left(\dfrac1{\sqrt2}\right)=\dfrac\pi4$

Now the surface area of the cap is given by the surface integral,

$\displaystyle\iint_{\text{cap}}\mathrm dS=\int_{\theta=0}^{\theta=2\pi}\int_{\varphi=0}^{\varphi=\pi/4}\|\mathbf r_u\times\mathbf r_v\|\,\mathrm dv\,\mathrm du$
$=\displaystyle\int_{u=0}^{u=2\pi}\int_{\varphi=0}^{\varphi=\pi/4}9\sin v\,\mathrm dv\,\mathrm du$
$=-18\pi\cos v\bigg|_{v=0}^{v=\pi/4}$
$=18\pi\left(1-\dfrac1{\sqrt2}\right)$
$=9(2-\sqrt2)\pi$

3 0

2 years ago

I need to turn this in in 2 hours please hurry

Nuetrik [128]

Answer:

2) 162°, 72°, 108°

3) 144°, 54°, 126°

Step-by-step explanation:

1) Multiply the equation by 2sin(θ) to get an equation that looks like ...

sin(θ) = <some numerical expression>

Use your knowledge of the sines of special angles to find two angles that have this sine value. (The attached table along with the relations discussed below will get you there.)

____

2, 3) You need to review the meaning of "supplement".

It is true that ...

sin(θ) = sin(θ+360°),

but it is also true that ...

sin(θ) = sin(180°-θ) . . . . the supplement of the angle

This latter relation is the one applicable to this question.

Similarly, it is true that ...

cos(θ) = -cos(θ+180°),

but it is also true that ...

cos(θ) = -cos(180°-θ) . . . . the supplement of the angle

As above, it is this latter relation that applies to problems 2 and 3.

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2 years ago