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3 years ago

Find the y and x intercept for 4x-y=12

Mathematics

1 answer:

aniked [119]3 years ago

4 0

Answer:

x intercept: (3,0)

y intercept: (0, -12)

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3 years ago

Find the volume of the region between the planes x plus y plus 2 z equals 2 and 4 x plus 4 y plus z equals 8 in the first octant

Alex787 [66]

Find the intercepts for both planes.

Plane 1, x + y + 2z = 2:

$y=z=0\implies x=2\implies (2,0,0)$

$x=z=0\implies y=2\implies(0,2,0)$

$x=y=0\implies 2z=2\implies z=1\implies(0,0,1)$

Plane 2, 4x + 4y + z = 8:

$y=z=0\implies4x=8\implies x=2\implies(2,0,0)$

$x=z=0\implies4y=8\impliesy=2\implies(0,2,0)$

$x=y=0\implies z=8\implies(0,0,8)$

Both planes share the same x- and y-intercepts, but the second plane's z-intercept is higher, so Plane 2 acts as the roof of the bounded region.

Meanwhile, in the (x, y)-plane where z = 0, we see the bounded region projects down to the triangle in the first quadrant with legs x = 0, y = 0, and x + y = 2, or y = 2 - x.

So the volume of the region is

$\displaystyle\int_0^2\int_0^{2-x}\int_{\frac{2-x-y}2}^{8-4x-4y}\mathrm dz\,\mathrm dy\,\mathrm dx=\displaystyle\int_0^2\int_0^{2-x}\left(8-4x-4y-\frac{2-x-y}2\right)\,\mathrm dy\,\mathrm dx$

$=\displaystyle\int_0^2\int_0^{2-x}\left(7-\frac72(x+y)\right)\,\mathrm dy\,\mathrm dx=\int_0^2\left(7(2-x)-\frac72x(2-x)-\frac74(2-x)^2\right)\,\mathrm dx$

$=\displaystyle\int_0^2\left(7-7x+\frac74 x^2\right)\,\mathrm dx=\boxed{\frac{14}3}$

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3 years ago

What is the slope of a line perpendicular to the line whose equation is 6x+8y=−128. Fully simplify your answer.

alexgriva [62]

The slope of a line perpendicular to the line 6x + 8y = -128 is 4/3.

Given,

The equation of a line = 6x + 8y = -128

We have to convert this into standard form of linear equation, y = mx + b :

So,

6x + 8y = -128

Add -6x to both sides,

6x + 8y - 6x = -6x - 128

Now,

8y = -6x - 128

Divide 8 on both sides

8y/8 = (-6x - 128) / 8

We get,

y = -6/8x - 16

Here, this is in standard form of linear equation.

Here slope of line, m₁ = -6/8

We have to find the slope of line(m₂) which is perpendicular to the given line.

If the line is perpendicular, the slope(m₂) will be the negative reciprocal of the slope(m₁) of the given line.

That is,

Slope of line, (m₂)= -(m₁) = - (-8/6) = 8/6 = 4/3

That is, the slope of the line which is perpendicular to the given line is 4/3.

Learn more about slope of the line here:

brainly.com/question/16616984

#SPJ1

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2 years ago