(9q r+6qr +20g)+(-6qr -qr -11q-4q r)=

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

Max wants to cover this rectangle with blue tiles I estimate that there would be

Sladkaya [172]

I have no idea. Where is the diagram.

8 0

2 years ago

Can someone help plz!

Verizon [17]

Answer:

Do you still need this one?

Step-by-step explanation:

$x\geq 1$

3 0

3 years ago

Y=3x²+3x+1,find the value of the discrimination

Rufina [12.5K]

Answer:

Step-by-step explanation:

a = 3

b = 3

c = 1

Discriminate = sqrt(b^2 - 4ac)

Discriminate = sqrt(9 - 4*3*1)

Discriminate = sqrt(9 - 12)

Discriminate = sqrt(-3)

What negative value means that there are no real roots and the y intercept = 1. You should always graph this kind of question, which I am doing for you. I use Desmos which is a very handy program.

You should notice that the minimum is above the x axis. That means there are no real roots, just as the discriminate tells you.

5 0

2 years ago

A supply company manufactures copy machines. The unit cost C (the cost in dollars to make each copy machine) depends on the numb

Ugo [173]

To answer this item, we take the differential of the equation and equate to zero.

C(x) = 0.8x² - 256x + 25939

Differentiation,
dC(x) = 1.6x - 256
dC(x) = 1.6x - 256 = 0

The value of x from the derived equation above is 160.

Thus, the number of machine to be made in order to minimize the cost should be 160.

4 0

3 years ago

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