Question

Waterslide fun!


(not really)..


A water ride at a local water park has a ride shaped like a cone that acts like a funnel whereby guests swirl around the cone until they drop through its center. There is one ride for adults and a similar, smaller version for children. If the adult ride has a radius of 32 feet and the child ride has a radius of 24 feet, what is the ratio between the volumes of each ride?
Answer
4:3
16:9
64:27
12:9

Answer 1

Do you know how to calculate the volume of a cone?

Answer 2

yes

Answer 3

So the ratio of the radii of the adult ride to the child's ride must be equal to the ratio of the height of the adult ride to the child's ride. The ratio of radii = 32:24=4:3. But they are asking for the ratio of the Volumes of each ride. What do you suppose that ratio would be?

Answer 4

hmm.. im not sure.. would we figure out the volume of both?

Answer 5

We don't have enough info to figure out the volume of both, we don't know the height of either. But we know that the ratio of the radii has to be the ratio of the heights as well (because the cones are similar). So the height of the adult ride is 4/3 the height of the child ride or 4/3 h.

Answer 6

ok.. so now what lol..

Answer 7

Let r=radius of kid ride, then radius of adult ride = (4/3)r
Let h=height of kid ride, then height of adult ride = (4/3)h
The volume of a cone is given by\[V=\frac13\pi r^2h\]So what is the ratio of the volume of the adult ride to the volume of the kid ride?

Answer 8

A?

Answer 9

You need to evaluate the volume of the adult over the volume of the kid
\[\frac{V _{a}}{V _{k}}=\frac{\frac13\pi r _{a}^2h _{a}}{\frac13\pi r _{k}^2h _{k}}=\frac{ r _{a}^2h _{a}}{ r _{k}^2h _{k}}\]but the (adult r) = (4/3)r and (adult h) = (4/3)h so\[=\frac{(\frac 43 r)^2h}{\frac 43 r^2h}\] Can you simplify this?

Answer 10

no?

Answer 11

Sorry. That last expression is wrong. It should have been\[=\frac{(\frac 43 r)^2\frac43 h}{\frac 43 r^2h}\] Can you simplify this?

Answer 12

i don't see a way to simplify..

Answer 13

Would it be easier if I presented it like this?\[=\left( \frac{(\frac43)^2\frac43}{\frac43} \right)\left( \frac{r^2}{r^2} \right)\left( \frac hh \right)\]

Answer 14

eh.. i still wouldn't know what to do.

Answer 15

What is the simplification of\[2\over 2\]

Answer 16

or how about \[x^2 \over x\]

Answer 17

1?

Answer 18

\[\frac 22=1\]\[\frac{x^2}{x}=x\]

Answer 19

So what is \[\frac{r^2}{r^2}=?\]\[\frac hh=?\]

Answer 20

And if you say 4/3=x, then what is \[\frac{x^3}{x}=?\]

Answer 21

4/3 h r^2

Answer 22

Not quite.\[\left( \frac{(\frac43)^2\cancel{\frac43}}{\cancel{\frac43}} \right)\left(\cancel{ \frac{r^2}{r^2}} \right)\left( \cancel{\frac hh} \right)=\left( \frac43 \right)^2\left( 1 \right)\left( 1 \right)=\frac{16}{9}\]

Answer 23

Oh. I just don't really understand it.. but thanks for your help.

Answer 24

Well, I tried to explain it but......hmmmm. Sorry.

Answer 25

np :/