Go to it! My answer will eventually appear in the comments, but right now I suspect the correct response is (D)-- cannot be determined. Why? Because there are four pieces of data we need to know, and we don't know any of them:
1. initial height
2. "diminished" height
3. initial radius
4. "diminished" radius
Still, that's just a conjecture at this point.* When I finally write out my attempted solution, I may disagree with what I've just written. Stay tuned-- but feel free to leave your answer in the comments if you think you've arrived at it before I have.
*If we think of the problem two-dimensionally, i.e., in terms of similar triangles, it may be that there is a solution. Again, I'm typing all this in a rush, so I really don't know yet. Will have to stare hard at the problem before I give it a go.
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1 comment:
I think the answer may in fact be (A): Quantity A is greater.
Because I can't use subscript and superscript fonts, I'm going to designate our four relevant variables this way:
H = the height of the full cone with 1000π units of water in it.
h = the height of the cone whose water volume has been reduced by 60%.
R = the radius of the larger cone.
r = the radius of the smaller cone.
The relationships between the heights and radii of the two cones can be expressed as:
r/R = h/H
Why? Think: similar triangles.
Imagine that we're looking at a cross-section of the two cones. It might look something like this:
__________ (H,R)
_________
________
_______
______
_____
____ (h,r)
___
__
_
--where (H,R) represents "height H, radius R," and (h,r) represents "height h, radius r." This cross-section represents two similar triangles: they share the same bottom angle, and the respective radii extend outward from the vertical edge at 90 degrees. That's enough to prove AAA (angle-angle-angle) similarity, which means the corresponding sides are proportionally related. So, once again:
r/R = h/H
The volumes of these two cones can be expressed this way:
V(sm) = (1/3)π(r^2)h
V(lg) = (1/3)π(R^2)H
We know that the volume of the smaller cone is only 40% that of the larger cone, so:
V(sm)/V(lg) = 400π/1000π = 400/1000
Let's leave that unreduced for the moment. You'll see why.
We can now set up a ratio:
V(sm)/V(lg) = 400/1000 =
[(1/3)π(r^2)h]/[(1/3)π(R^2)H]
Let's rewrite the above fraction in a more "vertical" form so you can see what's going to happen next.
(1/3)π(r^2)h
----------------
(1/3)π(R^2)H
We can immediately cancel out the [(1/3)π] on the top and bottom:
(r^2)h
----------
(R^2)H
...and now it's a good time to remember that r/R = h/H. This means that we can replace the r and R with h and H, and still preserve the ratio:
h^3/H^3 = 400/1000
So then we take the cube root of both sides (which I'll write using fractional powers since there's no cube-root symbol that I can use in the comment thread), and we get:
h/H = 400^(1/3)/10
So what's the cube root of 400? I can't say right away, but I do know it's a number greater than 4. In fact, I can reason that it's between 4 and 10: 4^3 = 64 and 10^3 = 1000. The cube root of 400 lies between these whole numbers. Since h/H is the very quantity we're comparing with 0.4, then the answer is clearly that Quantity A is greater.
(In case you're wondering, my calculator says the cube root of 400 is approximately 7.368063.)
QED.
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