Question

A metallic sphere is melted an then recasted into128 cones each of radius 4 cm and height 8 cm.Find the radius of the sphere.

Answer 1

@hartnn

Answer 2

@hartnn

Answer 3

@pitamar

Answer 4

Can you find the volume of a single cone?

Answer 5

\[\frac{ 1 }{3? }*\pi*r^2h\]

Answer 6

Yes

Answer 7

So now?

Answer 8

Now take this volume of a single cone and multiply by 128, because you have 128 cones. What do you get?

Answer 9

Okay,Lemme calculate.

Answer 10

How is it going?

Answer 11

134.04 times 128?

Answer 12

Do not approximate it just yet =)
The answer may include \(\pi\) and can be a fraction. Let's just form it.
If you want, I can show you what I mean.

Answer 13

Sure.

Answer 14

I made a mistake. let me fix it =|

Answer 15

We know we a single cone has radius of 4cm and height of 8cm:
$$\text{cone_volume} = \frac{\pi (4)^2 \cdot 8}{3} = \frac{\pi \big(2^2\big)^2 \cdot 8}{3} = \frac{\pi 2^4 \cdot 2^3}{3} = \frac{\pi 2^7}{3}$$
We have 128 cones in total so:
$$\text{cones_volume} = 128 \cdot \text{cone_volume} = 128 \cdot \frac{\pi 2^7}{3} = \\
= 2^7 \cdot \frac{\pi 2^7}{3} = \frac{\pi 2^{7 + 7}}{3} = \frac{\pi 2^{14}}{3}
$$Ok?

Answer 16

I didnt get the lst step.

Answer 17

The first step?

Answer 18

Do you understand why \(\big(2^2\big)^2 = 2^4\)?
What about \(2^4 \cdot 2^3 = 2^7\)?

Answer 19

last step.

Answer 20

128 is basically \(2^7\) so:
$$
128 \cdot \frac{\pi 2^7}{3} =\frac{\pi 2^7 \cdot 128}{3} = \frac{\pi 2^7 \cdot 2^7}{3} = \frac{\pi 2^{7+7}}{3} = 128 \cdot \frac{\pi 2^{14}}{3}
$$Which is very close to what happend with the \(2^4\) and \(2^3\) above.

Answer 21

We do this to find the volume of all the cones together. If we know the volume of a single cone and we know that we have 128 of these, then we just multiply the single-cone volume by 128 to get the total volume of all the cones.
Is it clear so far?

Answer 22

128 is used twice why?

Answer 23

copy paste mistake. Let me rewrite this. Look above you'll see I've only used it once =p

Answer 24

$$
128 \cdot \frac{\pi 2^7}{3} =\frac{\pi 2^7 \cdot 128}{3} = \frac{\pi 2^7 \cdot 2^7}{3} = \frac{\pi 2^{7+7}}{3} = \frac{\pi 2^{14}}{3}
$$

Answer 25

Here =)

Answer 26

so 128^2*pi/3?

Answer 27

yes, but I've written this as a power of 2 for a reason. So I prefer to stick with \(\frac{\pi 2^{14}}{3}\) for now, ok?

Answer 28

so
16384/3 * 22/7?

Answer 29

Well, again you try to approximate. Remember we are not asked about a volume or anything, we are asked about the radius of the sphere, which brings to the question "Why did we calculate the this volume then?".
Well, if we assume that the sphere is now hollow (because it isn't said to be in the question) and that we didn't somehow change the density of the material it is made of and made it take less volume.... then we can say that the melting process didn't change the total volume of what we have.
That means that the volume of our cones is the volume of our sphere. Do you know the volume formula for sphere?

Answer 30

\[=\Pi \times r^2 \times h\]

Answer 31

Well, sphere doesn't have any height so that cannot be true. Sphere only has radius. the formula is:
$$
\text{sphere_volume} = \frac{4\pi r^3}{3}
$$This has a neat calculus proof, if you know some calculus then maybe I can show you later =)
Anyway, we know that our sphere volume is equal to our total cones volume. So:
$$
\text{sphere_volume} = \text{cones_volume} \\
\frac{4\pi r^3}{3} = \frac{\pi 2^{14}}{3}
$$Ok?

Answer 32

n0w?

Answer 33

r^2 = 2^14?

Answer 34

Not exactly.
We can multiply both sides by 3 to get rid of it for example. What else can we get rid of on both sides?

Answer 35

pi

Answer 36

Right. and now we're left with:
$$
\frac{4\cancel \pi r^3}{\cancel3} = \frac{\cancel\pi 2^{14}}{\cancel3} \\
4 r^3 = 2^{14}
$$
We still have to get rid of the 4 on the left. So what is \(\frac{2^{14}}{4}\)?

Answer 37

16384/4?

Answer 38

Don't work so hard =) we can use power rules again in our favour
So we have say \(4 = 2^2\) which makes it \(\frac{2^{14}}{2^2}\)
What happens now?

Answer 39

2^12?

Answer 40

Exactly. So
$$ r^3 = 2^{12} $$
Now what?

Answer 41

4096?

Answer 42

so 64^2 is 4096

Answer 43

We have no need to change the form just yet. We are still in our way to find r and this form might actually help us.
What is the next operation we do in order to find r?

Answer 44

$$r^3 = 2^{12}$$

Answer 45

Well, you don't have to compute it, just name it =)
What is the last thing that we have to get rid of in order to find \(r\)?

Answer 46

pi

Answer 47

sorry sorry 4

Answer 48

We don't have pi anymore (or 4), we have
$$r^3 = 2^{12}$$
It's good thing that we know \(r^3\), but we want \(r\). What should we do to change the \(r^3\) to \(r\)?

Answer 49

Hint?

Answer 50

sqrt2^12

Answer 51

Well we're getting closer. square root (\(\sqrt{\text{something}}\)) will get rid of a power of 2, but we have a power of 3, so we need a cube root:
$$
r^3 = 2^{12} \\
\sqrt[3]{r^3} = \sqrt[3]{2^{12}} \\
r = \sqrt[3]{2^{12}}
$$

Answer 52

Now, if you remember power rules (again), then cube root is just like power of 1/3. So we can say:
$$
\sqrt[3]{2^{12}} = \big(2^{12}\big)^{\frac{1}{3}}
$$What happens now?

Answer 53

2^4

Answer 54

exactly. So r = 2^4 = ?

Answer 55

64 which i had said 15 min back.

Answer 56

Thanks anyways.

Answer 57

\(2^4 = 64\)?

Answer 58

16 sorry

Answer 59

Right =)
You're welcome