Question

A wooden pyramid, 3 inches tall, has a square base. A carpenter increases the dimensions of the wooden pyramid by a factor of 6 and makes a larger pyramid with the new dimensions. Describe in complete sentences the ratio of the volumes of the two pyramids.

Answer 1

Do you know the formula for calculating the volume of a square-based pyramid?

Answer 2

is it V=1/3*b^2*h

Answer 3

Right. If the carpenter increase the dimensions of the pyramid by a factor of 6, how would you rewrite the equation for the volume?

Answer 4

Close.\[V=\frac13(6b)^2*6h\]The ratio of the volumes of the two pyramids would be given by \[\left( V _{6}\over V _{1} \right)=\left( \frac13(6b)^2*6h \over \frac 13 b^2+h\right) \]

Answer 5

Can you simplify the ratio?

Answer 6

Sorry, I got that last equation wrong. It should have been\[\left( V _{6}\over V _{1} \right)=\left( \frac13(6b)^2*6h \over \frac 13 b^2*h\right)\]

Answer 7

o wow okay!...so whats the b? i think the h is 3?

Answer 8

You should simplify the expression. Do you know how?

Answer 9

I think i do but i cant remember right now!

Answer 10

Put like terms together. The terms that are exactly the same on the top and bottom would cancel. \[\left( V _{6}\over V _{1} \right)=\left( \frac13(6b)^2*6h \over \frac 13 b^2*h\right)=\left( \frac13 \over \frac 13\right)\left( 6b \over b \right)^2\left( 6h \over h \right)\]What would be left?

Answer 11

(1)(5b)^2(5h)??

Answer 12

Let's take one term\[\left( 6b \over b \right)^2=6^2\left( b \over b \right)^2=?\]

Answer 13

Okay. \[6^2\left( b \over b \right)^2=6^2(1)^2=6^2(1)=6^2\]

Answer 14

omg o okay i got you!

Answer 15

v=6^2*6?

Answer 16

Correct!! So the ratio would be\[\left( V _{6}\over V _{1} \right)=\left( \frac13 \over \frac 13\right)\left( 6b \over b \right)^2\left( 6h \over h \right)=(1)(6^2)(6)=6^3\]

Answer 17

So what sentence can you say about the ratios of the volumes of the two pyramids?

Answer 18

The volume larger pyramid (where the dimensions were 6 times as great) is ______times the volume of the smaller pyramid.

Answer 19

6 times?

Answer 20

Nope, 6 times 6 times 6.....or 6^3 or 216 times greater