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# The stiffness of a diving board is proportional to the cube

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The stiffness of a diving board is proportional to the cube [#permalink]  21 Jun 2017, 13:06
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60% (02:30) correct 39% (03:04) wrong based on 28 sessions

The stiffness of a diving board is proportional to the cube of its thickness and inversely proportional to the cube of its length. If diving board A is twice as long as diving board B and has 8 times the stiffness of diving board B, what is the ratio of the thickness of diving board A to that of diving board B? (Assume that the diving boards are equal in all respects other than thickness and length.)

[Reveal] Spoiler: OA
4

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Re: The stiffness of a diving board is proportional to the cube [#permalink]  25 Sep 2017, 07:37
1
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We can translate what is written in formulas as $$s=\frac{t^3}{l^3}$$ where s is stiffness, t is thickness and l is length. Then, if board A has 3 times the length and 8 times the stiffness of table B, the formula is rewritten as $$8s=\frac{t^3}{8*l^3}$$ so that $$t^3=8s*8l^3$$. Thus the ratio between the cubic thickness of the two tables is $$\frac{64sl^3}{sl^3}=64$$ but since this is the cubic ratio we have to compute the cubic root to find the ratio between thickness, i.e. $$64^{\frac{1}{3}}=4$$
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Re: The stiffness of a diving board is proportional to the cube [#permalink]  29 Nov 2017, 07:45
IlCreatore wrote:
We can translate what is written in formulas as $$s=\frac{t^3}{l^3}$$ where s is stiffness, t is thickness and l is length. Then, if board A has 3 times the length and 8 times the stiffness of table B, the formula is rewritten as $$8s=\frac{t^3}{8*l^3}$$ so that $$t^3=8s*8l^3$$. Thus the ratio between the cubic thickness of the two tables is $$\frac{64sl^3}{sl^3}=64$$ but since this is the cubic ratio we have to compute the cubic root to find the ratio between thickness, i.e. $$64^{\frac{1}{3}}=4$$

Can someone explain this to me? I am really struggling with this problem. I understand how you got to S = t^3/l*3, but I don't understand how you got to the next step. How did you end up with 8*l^3 in the denominator?

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Re: The stiffness of a diving board is proportional to the cube [#permalink]  29 Nov 2017, 10:01
I believe 6 would be the right answer.
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Re: The stiffness of a diving board is proportional to the cube [#permalink]  29 Nov 2017, 12:49
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Please, Guys, refering to the explanation provided by @ilcreatore. it is perfect.

The OA is 4.

Regards
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Re: The stiffness of a diving board is proportional to the cube [#permalink]  29 Nov 2017, 13:44
1
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The error is the incorrect transcription of the question. The initial question and math is done with the length being twice as great while the forum explanation is 3 times.
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Re: The stiffness of a diving board is proportional to the cube [#permalink]  27 May 2019, 05:06
We have the stiffness s of a board with thickness t and length l is given by s = t^3/l^3.
Thus, stiffness of board A with thickness tA and length lA i.e. SA = tA^3/lA^3.
Similarly, stiffness of board B with thickness tB and length lB i.e. SB = tB^3/lB^3.

Now we have board A is twice as long as board B and has 8 times the stiffness of board B.
Thus, lA = 2*lB and SA = 8*SB.
or, tA^3/lA^3 = 8*tB^3/lB^3
or, tA^3/(2*lB)^3 = 8*tB^3/lB^3
or, tA^3/8 = 8*tB^3
or, tA^3/tB^3 = 64
Therefore, tA/tB = 4
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Re: The stiffness of a diving board is proportional to the cube [#permalink]  27 May 2019, 08:08
1
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Expert's post
Quote:
The stiffness of a diving board is proportional to the cube of its thickness and inversely proportional to the cube of its length. If diving board A is twice as long as diving board B and has 8 times the stiffness of diving board B, what is the ratio of the thickness of diving board A to that of diving board B? (Assume that the diving boards are equal in all respects other than thickness and length.)

Remember that if an algebraic solution is proving difficult on the GRE, there is frequently an alternative tactic opportunity.

In this case, even though we have no answer choices to assist us, we can potentially plug in our own values to model the equation rather than setting up the algebra ourselves.

Since each of the measurements for Diving Board A are predicated on Diving Board B, plug in easily cubed values for the length and stiffness for Diving Board B first such as length B = 2 and stiffness B = 8.

Then according to the problem, since stiffness is proportional to the cube of the thickness and the inverse of the cube of the length, we can solve for the thickness B based on our values.

Therefore, 8 = thickness³ / length³ --> 8 = thickness³ / 2³ --> 64 = thickness³ --> thickness = 4.

Based on these values and according to the problem we know that length A = 2 x length B = 2 x 2 = 4, and that stiffness A = 8 x stiffness B = 8 x 8 = 64.

Now, follow the same process to solve for thickness A using our plugged in values.

So, 64 = thickness³ / length³ --> 64 = thickness³ / 4³ --> 4,096 = thickness³ --> thickness = 16.

Finally, set the sought ratio of thickness A / thickness B --> 16 / 4 = 4. Enter 4.
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Stefan Maisnier

Re: The stiffness of a diving board is proportional to the cube   [#permalink] 27 May 2019, 08:08
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