Question

A wooden pyramid, 9 inches tall, has a square base. A carpenter increases the dimensions of the wooden pyramid by a factor of 4 and makes a larger pyramid with the new dimensions. Describe in complete sentences the ratio of the volumes of the two pyramids.

Answer 1

@waterineyes

Answer 2

See here you are given with:

Height of Pyramid that is 9 in.

Now the formula for Volume of Pyramid is:

\[Volume = \frac{1}{3} \times l \times w \times h\]

Answer 3

okay, so what do I do?

Answer 4

And one more thing, square base means length is equal to width of that pyramid..

So, l= w.

\[Volume = \frac{1}{3} \times 9 \times l^2\]

Answer 5

cuz I don't have the rest of the equation

Answer 6

so I just solve for that?

Answer 7

Can you simplify the above that I gave you??

Keep l^2 as it is there, just divide 9 by 3 there, what you got?

Answer 8

volume = 9 * l^2

Answer 9

9/3 = ??

Answer 10

oh crap sorry im meant to put 3

Answer 11

volume = 3 * l^2

Answer 12

Yep..

So this is the original volume, let us say it as 1, means at base of volume and l, just put 1 like this:

\[\large Volume_1 = 3 \times l^2_1\]

Getting?

Answer 13

Idk :0

Answer 14

Just put 1 in the base to show that this is the volume of first pyramid, because now pyramid has been modified by changing its base..

Answer 15

so... volume = 3 * 1???

Answer 16

In simple language, if you have two pyramids, then how will you differentiate their volumes??

You will let \(V_1\) as volume of first pyramid and you will suppose \(V_2\) as volume of second pyramid.

Right?

Answer 17

yes

Answer 18

Hey, \(Volume_1\) and \(l_1\) is just showing that these are for first pyramid.

Similarly now we will find for second case..

Answer 19

Now, in question the square base has been changed to 4 times .

Means l, w and h are increased by factor of 4..

So:

\[l_2 = 4l_!\]
\[w_2 = 4w_1\]
\[h_2 = 4 h_1\]

Answer 20

okay yu are confusing me

Answer 21

ALOT

Answer 22

What I think, you always confuse between why we take 1 and 2 in bases..

Answer 23

es I don't understand the sub 1 and 2

Answer 24

*yes

Answer 25

Hey they are just Volume, length, width and height but sub in 1 shows that they are for first pyramid..

Now what happen when I will changed their length, width and height, automatically volume will change now..

So I let Volume, length, width , height again but with different base ie 2 because now the previous pyramid has changed and we have one newer and second pyramid..

Answer 26

And one thing you will not agree to me, but I don't know how to draw pyramid, so I can't make you understand practically by doing drawings here..

Answer 27

okay so the 1 and 2 represent pyramid 1 and pyramid 2. right?

Answer 28

Yep...

Similarly we have two pyramid in our question..

Now can I let \(V_1, l_1, w_1, h_1\) for first pyramid?

Answer 29

okay I understand

Answer 30

So,

In first pyramid you are given with:

Volume = \(V_1\)
\(l_1 = w_1\)
\(h_1 = 9\)

Getting this much?

Answer 31

yes

Answer 32

So just apply the formula here for Volume:

\[Volume = \frac{1}{3} \times length \times width \times height\]
\[V_1 = \frac{1}{3} \times l_1 \times w_1 \times h_1\]
\[V_1 = \frac{1}{3} \times l_1 \times w_1 \times 9\]

Getting?

Answer 33

Here \(w_1 = l_1\), so can I write \(w_1\) as \(l_1\)??

Answer 34

\[V _{1} = 3 \times l _{1} \times w\]

Answer 35

* \[w _{1}\]

Answer 36

But \(w_1 = l_1\)

Right?

Answer 37

is that right?

Answer 38

yes

Answer 39

Yep that is right but can be changed further..

Answer 40

so can you just put\[l _{1}^{2}\]

Answer 41

So tell me:

\[V_1 =3 \times l_1 \times l_1\]

Right?

Answer 42

yes

Answer 43

Yep..

So you got:

\[V_1 = 3 l^2_1\]

Remember this.. Now we will move to the second pyramid..

Can I let \(V_2, l_2, w_2, h_2\) for second pyramid now?

Answer 44

okay go it

Answer 45

*got

Answer 46

So:

Now you are given with that the length, height and width of the earlier that is first pyramid are changed and they are each multiplied by 4 to make new pyramid..

Getting?

Answer 47

Meaning \(l_1, h_1, w_1\) all are multiplied by 4 to get new pyramid.

Answer 48

\[V _{2} =12 \times l _{2}^{6} ??\]

Answer 49

Good try but no..

Answer 50

:(

Answer 51

I mean now:

\[l_2 = 4 l_1\]
\(w_2 = 4w_1\)
\(h_2 = 4h_1\)
Use this to find \(V_2\).. :)

Answer 52

ohmygosh. im on a timed practice test and my time is almost up!

Answer 53

Sorry I can't do anything in it..

Answer 54

\[V _{2} = 4l _{1} \times 4w _{1} \times 4h _{1} ?????\]

Answer 55

You are missing 1/3 there..

Answer 56

ok I give up. thx for trying to help

Answer 57

You need to be more strong.

Don't ever give up..

You were just near the answer..

Answer 58

Thanks for tolerating me this long.. :)

Answer 59

no im not! I have been working on this one problem for over an hour nd I still don't get it. im just getting more and more confused

Answer 60

haha thanks for tolerating me