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3 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]3 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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What is the y-intercept of a line that has a slope of 3 and passes through point (–1, –7)?

Stells [14]

Answer:

y-intercept is -4

Step-by-step explanation:

(Refer to image)

Substitute the point and slope with the slope formula: y=mx+b and simplify

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2 years ago

Determined to find the slope<br>(1 7K), and (-3,5K)

Helga [31]

Answer: $\dfrac{K}{2}$ .

Step-by-step explanation:

As we know ,

The slope of a line that passes through $(x_1,y_1)$ and $(x_2,y_2)$ is given by :

$\dfrac{y_2-y_1}{x_2-x_1}$

The slope of a line that passes through (1 , 7K), and (-3,5K) =

$\dfrac{5K-7K}{-3-1}\\\\=\dfrac{-2K}{-4}\\\\=\dfrac{K}{2}$

Hence, the slope of the given line is $\dfrac{K}{2}$ .

8 0

3 years ago

Find the equation of the sphere if one of its diameters has endpoints (4, 2, -9) and (6, 6, -3) which has been normalized so tha

Pavel [41]

Answer:

$(x - 5)^2 + (y - 4)^2 + (z - 6)^2 = 14$ .

(Expand to obtain an equivalent expression for the sphere: $x^2 - 10\,x + y^2 - 8\, y + z^2 - 12\, z + 63 = 0$ )

Step-by-step explanation:

Apply the Pythagorean Theorem to find the distance between these two endpoints:

$\begin{aligned}&\text{Distance}\cr &= \sqrt{\left(x_2 - x_1\right)^2 + \left(y_2 - y_1\right)^2 + \left(z_2 - z_1\right)^2} \cr &= \sqrt{(6 - 4)^2 + (6 - 2)^2 + ((-3) - (-9))^2 \cr &= \sqrt{56}}\end{aligned}$ .

Since the two endpoints form a diameter of the sphere, the distance between them would be equal to the diameter of the sphere. The radius of a sphere is one-half of its diameter. In this case, that would be equal to:

$\begin{aligned} r &= \frac{1}{2} \, \sqrt{56} \cr &= \sqrt{\left(\frac{1}{2}\right)^2 \times 56} \cr &= \sqrt{\frac{1}{4} \times 56} \cr &= \sqrt{14} \end{aligned}$ .

In a sphere, the midpoint of every diameter would be the center of the sphere. Each component of the midpoint of a segment (such as the diameter in this question) is equal to the arithmetic mean of that component of the two endpoints. In other words, the midpoint of a segment between $\left(x_1, \, y_1, \, z_1\right)$ and $\left(x_2, \, y_2, \, z_2\right)$ would be:

$\displaystyle \left(\frac{x_1 + x_2}{2},\, \frac{y_1 + y_2}{2}, \, \frac{z_1 + z_2}{2}\right)$ .

In this case, the midpoint of the diameter, which is the same as the center of the sphere, would be at:

$\begin{aligned}&\left(\frac{x_1 + x_2}{2},\, \frac{y_1 + y_2}{2}, \, \frac{z_1 + z_2}{2}\right) \cr &= \left(\frac{4 + 6}{2},\, \frac{2 + 6}{2}, \, \frac{(-9) + (-3)}{2}\right) \cr &= (5,\, 4\, -6)\end{aligned}$ .

The equation for a sphere of radius $r$ and center $\left(x_0,\, y_0,\, z_0\right)$ would be:

$\left(x - x_0\right)^2 + \left(y - y_0\right)^2 + \left(z - z_0\right)^2 = r^2$ .

In this case, the equation would be:

$\left(x - 5\right)^2 + \left(y - 4\right)^2 + \left(z - (-6)\right)^2 = \left(\sqrt{56}\right)^2$ .

Simplify to obtain:

$\left(x - 5\right)^2 + \left(y - 4\right)^2 + \left(z + 6\right)^2 = 56$ .

Expand the squares and simplify to obtain:

$x^2 - 10\,x + y^2 - 8\, y + z^2 - 12\, z + 63 = 0$ .

8 0

3 years ago

Suppose a basketball player has made 390 out of 434 free throws. If the player makes the next 2 free throws, I will pay you $40.

alukav5142 [94]

Answer: a) -$0.19, b) -$111.72 .

Step-by-step explanation:

Since we have given that

Number of free throws = 434

Number of throws made by them = 390

Amount for making the next 2 free throws = $40

Amount otherwise he has to pay = $169

a) Find the expected value of the proposition.

Expected value of success in next 2 free throws = $\dfrac{390}{434}\times \dfrac{391}{435}=0.8077$

Expected value would be

$0.8077\times 40+(1-0.8077)\times -169\\\\=32.308-32.4987\\\\=-\$0.19$

b) If you played this game 588 times how much would you expect to win or lose?

Number of times they played the game = 588

So, Expected value would be

$588\times -0.19\\\\=-\$111.72$

Hence, a) -$0.19, b) -$111.72

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3 years ago

How to determine whether to move right or left on the number line when adding rational numbers?

Vedmedyk [2.9K]

So if you're adding two negatives you move left, positive you move right, and when you have to determine a negative, and a positive you just have to see which one is the farthest from zero which the farthest one's sign is either positive, or negative. Positive to move right, and negative to move left.

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3 years ago