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3 years ago

This is all first in expanded form then standard form

Mathematics

1 answer:

alex41 [277]3 years ago

6 0

5^4
11^2
2^8
12^3 not sure if you needed the answers to these individually but if so lmk

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Find the standard equation of a sphere that has diameter with the end points given below. (3,-2,4) (7,12,4)

DiKsa [7]

Answer:

The standard equation of the sphere is $(x-5)^{2} + (y-5)^{2} + (z-4)^{2} = 53$

Step-by-step explanation:

From the question, the end point are (3,-2,4) and (7,12,4)

Since we know the end points of the diameter, we can determine the center (midpoint of the two end points) of the sphere.

The midpoint can be calculated thus

Midpoint = $(\frac{x_{1} + x_{2} }{2}, \frac{y_{1} + y_{2} }{2}, \frac{z_{1} + z_{2} }{2})$

Let the first endpoint be represented as $(x_{1}, y_{1}, z_{1})$ and the second endpoint be $(x_{2}, y_{2}, z_{2})$ .

Hence,

Midpoint = $(\frac{x_{1} + x_{2} }{2}, \frac{y_{1} + y_{2} }{2}, \frac{z_{1} + z_{2} }{2})$

Midpoint = $(\frac{3 + 7 }{2}, \frac{-2+12 }{2}, \frac{4 + 4 }{2})$

Midpoint = $(\frac{10 }{2}, \frac{10}{2}, \frac{8 }{2})\\$

Midpoint = $(5, 5, 4)$

This is the center of the sphere.

Now, we will determine the distance (diameter) of the sphere

The distance is given by

$d = \sqrt{(x_{2} - x_{1})^{2} +(y_{2} - y_{1})^{2} + (z_{2}- z_{1})^{2} }$

$d = \sqrt{(7 - 3)^{2} +(12 - -2)^{2} + (4- 4)^{2}$

$d = \sqrt{(4)^{2} +(14)^{2} + (0)^{2}$

$d = \sqrt{16 +196 + 0$

$d =\sqrt{212}$

$d = 2\sqrt{53}$

This is the diameter

To find the radius, r

From $Radius = \frac{Diameter}{2}$

$Radius = \frac{2\sqrt{53} }{2}$

∴ $Radius = \sqrt{53}$

r = $\sqrt{53}$

Now, we can write the standard equation of the sphere since we know the center and the radius

Center of the sphere is $(5, 5, 4)$

Radius of the sphere is $\sqrt{53}$

The equation of a sphere of radius r and center $(h,k,l)$ is given by

$(x-h)^{2} + (y-k)^{2} + (z-l)^{2} = r^{2}$

Hence, the equation of the sphere of radius $\sqrt{53}$ and center $(5, 5, 4)$ is

$(x-5)^{2} + (y-5)^{2} + (z-4)^{2} = \sqrt{(53} )^{2}$

$(x-5)^{2} + (y-5)^{2} + (z-4)^{2} = 53$

This is the standard equation of the sphere

6 0

3 years ago

How would a find a rule for this cubic function?

ELEN [110]

Hello here is a solution :

3 0

3 years ago

According to the recent poll, 22 1/4% of the people polled said that they approved of the city plan to expand the library. If 10

Sphinxa [80]

Answer:

Some 224 people were in favor of the expansion.

Step-by-step explanation:

Given that, according to the recent poll, 22 1/4% of the people polled said that they approved of the city plan to expand the library, if 1004 people was polled, to determine how many were in favor of the expansion should be done the following calculation:

1/4 = 0.25

100 = 1004

22.25 = X

1004 x 22.25 / 100 = X

223.39 = X

Thus, some 224 people were in favor of the expansion.

3 0

3 years ago

Convert a recipe that serves 6 and calls for 3/4 cups of sugar to one that serves 20.

djyliett [7]

Answer:

5/2cups

Step-by-step explanation:

If a recipe serves 6 and calls for 3/4cups, arecipe that serves 20, will call for (20×3/4)/6 cups. Mathematically;

6recipe = 3/4cups

20recipe = x

Cross multiplying we have;

6x = 20×3/4

6x = 15

x = 15/6

x = 5/2

This means a recipe that serves 20 will call for 5/2cups

3 0

3 years ago

Consider the figure below. Find the measures of angle CRN and angle GMB

Katena32 [7]

Answer: Have you figured out the answered cuz I need it to please?

Step-by-step explanation:

3 0

3 years ago

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