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2 years ago

The city library is having a book sale to make room for the new books it purchased. The graph shows the cost, y, in dollars, to

Mathematics

2 answers:

Agata [3.3K]2 years ago

7 0

Answer:

Step-by-step explanation:

Zolol [24]2 years ago

4 0

Answer:

The

cost of the books

depends on the

(number of books purchased)

At the sale, each book costs $

(0.50)

Step-by-step explanation:

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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

2 years ago

Help me please with explanation

8_murik_8 [283]

Step-by-step explanation:

angle AOB = x+3

angle AOC = 2x + 11

angle BOC = 4x-7

angle AOC = angle AOB + angle BOC

=> 2x +11 = (x+3) + (4x-7)

2x +11 = 5x - 4

=> 3x = 15

x = 5

subst x = 5 in the given formulas

angle AOB = x +3 =8

angle AOC = 2x + 11 = 21

angle BOC = 4x - 7 = 13

7 0

3 years ago

2.4 Assignment 4

brilliants [131]

Answer:

Step-by-step explanation:

8 0

1 year ago

Solve for y in the following equation. x+2y=z

Galina-37 [17]

Answer:

The Answer:

Step-by-step explanation:

x+2y=z

minus x both sides

2y=z-x

divide by 2 both sides

The Answer:

$y = \frac{z - x}{2}$

6 0

3 years ago

Translate (3,2) to the left one unit. Then reflect the result over the y-axis. What are the coordinates of the final point?

OleMash [197]

Answer:

(-2, 2)

Step-by-step explanation:

Take point (3, 2) and first translate it left one unit. Now the point is (2, 2)

Now reflect it over the y-axis. Doing so keeps the y value the same, we have a vertical line of reflection, so we are only changing the distance to the y-axis, which is the x-value.

Our x value is 2 units away from the y axis, so its reflection will be 2 units away on the other side of the axis.

The new x value is -2, so the point is (-2, 2)

4 0

2 years ago