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2 years ago

What is the slope of a line perpendicular to the line whose equation is 12x-9y=135

Mathematics

1 answer:

trasher [3.6K]2 years ago

7 0

Answer:

$\frac{-9}{12}$

Step-by-step explanation:

First, we need to isolate y.

1. Subtract 12x from both sides.

-9y=-12x+135

2. Divide by -9 in order to isolate y.

$y=\frac{12}{9} x-15$

3. Understand that perpendicular equations' slopes are opposite and reciprocal to the original equation. The slope of the original equation is $\frac{12}{9}$ , so the reciprocal and opposite (flip and change the sign) slope would be $\frac{-9}{12}$ .

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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

3 years ago

Elsa's magic wand can cast 53 spells. Her wand can cast 17fewer spells than her sister Anna's wand, which is a little bigger and

Mademuasel [1]

Answer:

Anna's wand can cast 70 spells

Step-by-step explanation:

Elsa's wand can cast 53 spells

her wand casts 17 fewer cells than her sister Anna's

Amount of spells cast by Anna's wand = ?

We write the question down in the form of an equation

$x - 17 = 53$

where $x$ is the amount of spells Anna's wand can cast.

we then proceed to solve by collecting like terms to different sides of the equation. We'll have

$x = 53 + 17$

which leaves us with

$x = 70$

This means that Anna's wand can cast 70 spells.

4 0

3 years ago

I need help please.

sweet-ann [11.9K]

Answer:

Third Option: $y = \frac{5}{4}x$

Step-by-step explanation:

Given the points on the graph, (4, 5) and (-4, -5):

In order to determine the equation of the given graph in slope-intercept form, y = mx + b:

Use the given points to solve for the slope:

Let (x₁, y₁) = (-4, -5)

(x₂, y₂) = (4, 5)

m = (y₂ - y₁)/(x₂ - x₁)

$m = \frac{5 - (5)}{4 - (-4)} = \frac{5 + 5}{4 + 4} = \frac{10}{8} = \frac{5}{4}$

Therefore, the slope of the line is: $m = \frac{5}{4}$ .

Next, use one of the given points on the graph, (4, 5) to solve for the y-intercept, b:

y = mx + b

5 = $\frac{5}{4} (4)$ + b

5 = 5 + b

5 - 5 = 5 - 5 + b

0 = b

Therefore, the linear equation in slope-intercept form is: $y = \frac{5}{4}x$ . The correct answer is Option 3.

8 0

2 years ago

Find the value of X please help me its due in an hour

spayn [35]

Answer:

Step-by-step explanation:

4 0

2 years ago

I do not understand these.

Vladimir [108]

Answer:

k(-1) = 7.

Step-by-step explanation:

k(-1) is the value of the function k when x = -1.

You read this off the graph.

Find the value of the point on the graph which corresponds to x = -1. From x = -1 on the horizontal axis move straight up until you meet the graph of k. Now look horizontally to the right and read off the value on the vertical

y-axis

You'll find it is 7.

6 0

3 years ago