[ASK] Two Cones

Monox D. I-Fly · 2018-07-04 19:21:46

Azis wants to make two cones using cartons. The surface area of the first cone is twice the second. The side length of the first cone is also twice the second. Determine the ratio of those cones' radius!

s1 = 2s2
L1 = 2L2
πr1(r1 + s1) = 2πr2(r2 + s2)
r1(r1 + 2s2) = 2r2(r2 + s2)

I was stuck with quadratic equations...

Grantingriver · 2018-07-05 04:49:30

It is very simple! the surface areas of the first and the second cones are given, respectively, by:

and since from the givens, we have:

Therefore we get:

so the ratio is 1. I hope that answer your question.

Monox D. I-Fly · 2018-07-05 12:55:07

But we must take into account the area of the base.

Bob · 2018-07-05 20:07:26

hi Monox D. I-Fly

But we must take into account the area of the base.

Are you sure because I am also stuck with the same equations as you. There are 4 unknowns, L1, L2, r1, and r2. And only two constraints. That's not enough to determine the ratio of r1 to r2.

What is meant by 'cartons' in this question. If I was trying to make a cone I'd use a sheet of card. Maybe there's a clue there to a third constraint.

Bob

Monox D. I-Fly · 2018-07-05 20:15:23

In my language it said "luas permukaan", and it strictly include the base area. Otherwise they would use the phrase "luas selimut" (side area) instead.

Also, "cartons" is the kind of paper usually used in drawing books.

Bob · 2018-07-06 00:26:26

hi Monox D. I-Fly

Ok. Still puzzled. I thought I'd try some numbers. I fixed L1 at 10 and L2 at 5.

I chose any number for r1 and tried r2 = 5,4,3,2,and 1

In Excel there is a feature called 'goal seek'. You specify a cell and a target value for that cell, and choose a cell that may be varied to achieve the target.

So I computed the two areas and also area1 minus 2 times area2. The target is to make this value zero. When it is, area1 is twice area2. I chose to vary the r1 value to achieve this target. Here are the results (sorry the table is a bit lopsided. It was Ok when I copied it.)

L1 L2    r1    r2 area1    area2    A1-2*A2    ratio
10 5 6.180339868    5 314.159264 157.0796327 -1.34787E-06 1.236067974
10 5 4.848861716    4 226.1949133 113.0973355 0.000242224 1.212215429
10 5 3.544016352 3 150.7971241 75.39822369 0.000676761 1.181338784
10 5 2.280109903    2 87.96459494 43.98229715 6.38796E-07 1.140054952
10 5 1.082762919    1 37.69912669 18.84955592 1.48466E-05 1.082762919

The target, A1-2*A2 is not quite zero because the goal seek feature is only approximately accurate, but I think good enough to demonstrate my point. With different r1 and r2 values the area constraint is satisfied without the ratio being the same each time. So there is no unique answer.

Bob

Monox D. I-Fly · 2018-07-06 12:45:39

I see, not fit for 9 graders at all...

Grantingriver · 2018-07-08 11:36:47

It can fit to the 9 graders, just change the "total surface area" to the "lateral surface area" and everything will be fine. In fact it is a common practice in mathematics to refer to the lateral surface area as the "surface area" without any restriction. If the problem composer want the reader to include the areas of the bases they usually refer to that by the phrase "total surface area".

Monox D. I-Fly · 2018-07-08 14:28:10

Okay, thanks.

Math Is Fun Forum

#1 2018-07-04 19:21:46

[ASK] Two Cones

#2 2018-07-05 04:49:30

Re: [ASK] Two Cones

#3 2018-07-05 12:55:07

Re: [ASK] Two Cones

#4 2018-07-05 20:07:26

Re: [ASK] Two Cones

#5 2018-07-05 20:15:23

Re: [ASK] Two Cones

#6 2018-07-06 00:26:26

Re: [ASK] Two Cones

#7 2018-07-06 12:45:39

Re: [ASK] Two Cones

#8 2018-07-08 11:36:47

Re: [ASK] Two Cones

#9 2018-07-08 14:28:10

Re: [ASK] Two Cones

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