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2 years ago

Find the vector equation for the line of intersection of the planes 5x+3y−2z=−45x+3y−2z=−4 and 5x+5z=0

Mathematics

1 answer:

olga_2 [115]2 years ago

8 0

$5x+5z=0\implies x+z=0\implies z=-x$

$5x+3y-2z=7x+3y-2x-2z=7x+3y-2=-4\implies7x+3y=-2$

Letting $x=t$ and $z=-t$ , we have

$7t+3y=-2\implies y=-\dfrac73t-\dfrac23$

so the intersection is given by

$\mathbf r(t)=\left(t,-\dfrac73t-\dfrac23,-t\right)$

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2 years ago

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omeli [17]

Given:

The volume of the rectangular prism is

$b^{3}+8 b^{2} +19b+12$ ,

the height is h=(b+3)

1. The volume of a rectangular prism is (base area)*height

also, notice that the volume is a third degree polynomial, the height is a 1st degree polynomial, so the base area must be a 2nd degree polynomial, whose coefficients we don't know yet.
Let this quadratic polynomial be $(mb^{2}+nb+k)$

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$b^{3}+8 b^{2} +19b+12=(mb^{2}+nb+k)*(b+3)$

notice that $b^{3}$ is the product of the largest 2 terms: $mb^{2}$ and b, so m must be 1

also, notice that 12 is the product of the constants, k and 3

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we write the above equality again:

$b^{3}+8 b^{2} +19b+12=(b^{2}+nb+4)*(b+3)$

$(b^{2}+nb+4)(b+3)= b^{3}+3 b^{2} +nb^{2}+3nb+4b+12$

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now compare the coefficient with the left side:

$8 b^{2}=(n+3)b^{2}$

8=n+3

n=5

substituting n=5:

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Answer: $b^{2}+5b+4$

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2 years ago

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Answer:

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3 years ago

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