Question

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 33 feet and the child ride has a radius of 22 feet, what is the ratio between the volumes of each ride?

27:8

9:6

9:4

3:2

Answer 1

@jim_thompson5910 can you please help me?

Answer 2

The two cones are similar. So we know that

r1/r2 = k

for some constant k

In this case, r1 = 33 and r2 = 22, so


r1/r2 = k

33/22 = k

3/2 = k

k = 3/2


This means that we're multiplying the smaller radius (22 ft) by 3/2 or 1.5 to get the larger radius (33 ft)


Since the two cones are similar, the larger height (h1) is found by multiplying the smaller height (h2) by 1.5


So h2 = 1.5*h1


Now turn to the formula of a cone


V = (pi*r^2*h)/3


and then plug in r = 33 and h = x to get


V = (pi*33^2*x)/3

V = (1089pi*x)/3

V = 363pi*x


So the volume of the larger cone is exactly 363pi*x cubic feet.


Now let's plug in r = 22 and h = y to find the volume of the smaller cone

V = (pi*r^2*h)/3

V = (pi*22^2*y)/3

V = (484pi*y)/3


So the volume of the smaller cone is exactly (484pi*y)/3 cubic feet.


Since the larger height is 1.5 times the smaller height, we can say

x = 1.5y

which means

363pi*x

turns into

363pi*1.5y

544.5pi*y


So the volume of the larger cone is also exactly 544.5pi*y cubic feet


Now divide the two volumes (larger into smaller)


(Larger Volume)/(Smaller Volume)

(544.5pi*y)/((484pi*y)/3)

3.375


Since 3.375 converts to the improper fraction 27/8, this means that the ratio is 27:8

Answer 3

That seems like a lot of work...but...it leads you to the very useful shortcut


if the ratio of two corresponding lengths is a:b, then the ratio of the volumes is a^3:b^3

Answer 4

wow thanks and you actually explained it to where i understand. thanks @jim_thompson5910 :)

Answer 5

That's great. I'm glad I could help.