Find the values of the variables and lengths of the sides of this kite.<br> Y-3, x+5, 2x+5, x+10

The radii of Earth and Pluto are 6,371 kilometers and 1,161 kilometers, respectively. Approximately how many spheres the size of

antoniya [11.8K]

The volume of a sphere is (4/3) (pi) (radius cubed).

The volume of one sphere divided by the volume of another one is

(4/3) (pi) (radius-A)³ / (4/3) (pi) (radius-B)³

Divide top and bottom by (4/3) (pi) and you have (radius-A)³ / (radius-B)³
and that's exactly the same as
( radius-A / radius-B ) cubed.

I went through all of that to show you that the ratio of the volumes of two spheres
is the cube of the ratio of their radii.

Earth radius = 6,371 km
Pluto radius = 1,161 km

Ratio of their radii = (6,371 km) / (1,161 km)

Ratio of their volumes = ( 6,371 / 1,161 ) cubed = about <u>165.2</u>

Note:
I don't like the language of the question where it asks "How many spheres...".
This seems to be asking how many solid cue balls the size of Pluto could be
packed into a shell the size of the Earth, and that's not a simple solution.
The solution I have here is simply the ratio of volumes ... how many Plutos
can fit into a hollow Earth if the Plutos are melted and poured into the shell.
That's a different question, and a lot easier than dealing with solid cue balls.

4 0

3 years ago

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Find the measure indicated. Assume that the lines which appear to be tangent are tangent. PART 3

ludmilkaskok [199]

I'll do problems 13 and 14 to get you started.

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Problem 13

<h3>Answer: 32 degrees</h3>

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Explanation:

Refer to the diagram below for problem 13. Since BD is a tangent of the circle, this means angle ABD is 90 degrees, and furthermore x+y = 90. This solves to y = 90-x.

If we focus on triangle ABC, then we can add the interior angles and set the sum equal to 180

A+B+C = 180

z+y+y = 180

z+2y = 180

z+2(90-x) = 180 .... plug in y = 90-x

z+180-2x = 180

z-2x = 180-180

z-2x = 0

z = 2x

We'll use this later.

Now move onto triangle ABD. This is a right triangle due to angle ABD being 90 degrees. The acute angles z and 26 are complementary, meaning they add to 90. So,

z+26 = 90

z = 90-26

z = 64

Now plug this into z = 2x and solve for x

z = 2x

64 = 2x

2x = 64

x = 64/2

x = 32

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Problem 14

<h3>Answer: 31 degrees</h3>

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Explanation:

Refer to the image shown below for problem 14. Like before, I've added various labels to it to help find the angle we're after (the red angle x).

Note how I've drawn a dashed line from point A to point D. This splits up quadrilateral ABCD into two triangles ABD and ACD. These triangles can be shown to be congruent using the HL (hypotenuse leg) theorem.

The congruent triangles also implies that angle DBC and angle BCD are the same measure (both are shown in green as the variable y).

Focus on triangle BCD. It has interior angles of: y, y and 62. Add them up, set the sum equal to 180.

y+y+62 = 180

2y+62 = 180

2y = 180-62

2y = 118

y = 118/2

y = 59

As the image attachment mentions, the angles x and y add to 90 degrees. This is because angle ABD is 90 degrees, due to segment BD being tangent to the circle. So x+y = 90 leads to x = 90-y, and therefore,

x = 90-y

x = 90-59

x = 31

Side note: it is not a coincidence that we ended up with half of the value of 62.