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3 years ago

Find the slope of (5,-5) and (-4,5)

Mathematics

1 answer:

Bas_tet [7]3 years ago

4 0

Answer:

Slope of points (5,-5) and (-4,5) is $\mathbf{-\frac{10}{9} }$

Step-by-step explanation:

We need to find the slope of points (5,-5) and (-4,5)

The formula used to find slope is: $Slope=\frac{y_2-y_1}{x_2-x_1}$

We have $x_1=5, y_1=-5, x_2=-4 \ and \ y=5$

Putting values in formula and finding slope

$Slope=\frac{y_2-y_1}{x_2-x_1}$

$Slope=\frac{5-(-5)}{-4-(5)}\\Slope=\frac{5+5}{-4-5}\\Slope=\frac{10}{-9}\\Slope=-\frac{10}{9}$

So, slope of points (5,-5) and (-4,5) is $\mathbf{-\frac{10}{9} }$

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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

What is the product 3x5(2x2+4x+1)

lesya692 [45]

Answer:

$6x^7+12x^6+3x^5$

Step-by-step explanation:

The given expression is:

$3x^5(2x^2+4x+1)$

We expand the parenthesis using the distributive property to obtain;

$3x^5(2x^2)+3x^5(4x)+3x^5(1)$

Recall that;

$a^m\imes a^n=a^{m+n}$

We apply this property and multiply out the constant terms to obtain;

$6x^7+12x^6+3x^5$

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4 years ago

What is the equation of the graph ?

ZanzabumX [31]

Answer:

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Step-by-step explanation:

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3 years ago

Solve the equation 4 1/5+b= 9 3/5

victus00 [196]

4 1/5 + b = 9 3/5

4 1/5 - 4 1/5 + b = 9 3/5 - 4 1/5
(taking the same amount from each side to keep it balanced)

0 + b = 5 2/5
(subtracting 4 1/5 from each side)

b = 5 2/5

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3 years ago

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After 9 new members joined the ski club, there were 38 members. How many members had been in the club previously?

dezoksy [38]

Answer:

Previously, there were 29 members in the ski club

Step-by-step explanation:

Subtract 9 from 38

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3 years ago

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