I believe 6 would be the right answer.

Please, Guys, refering to the explanation provided by @ilcreatore. it is perfect.

The OA is 4.

Regards

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.

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

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.

Ultimately we need to express the stiffness of diving board A in terms of board B

The first step is to derive the formula for stiffness. We know that the stiffness (S) of a diving board is proportional to the cube of its thickness (T); so \(S=T^3\). We also know that the stiffness of a diving board is inversely proportional to the cube of its length (L); so \(S=1/L^3\).

Putting these together you get: \(S=\frac{T^3}{L^3}\).

Now if we go back and re-read the question, we know that diving board A is twice as long as diving board B and has 8 times the stiffness of diving board B. To make things simple, let's say that the stiffness of diving board A is 8, and the length of diving board A is 2. This means that the stiffness of diving board B is 1 and its length is also 1.

Next, we need to determine the ratio of the thickness of diving board A to the thickness of diving board B. Keep in mind that the formula you created for stiffness needs to be re-written so that it solves for thickness cubed. So we change \(S=\frac{T^3}{L^3}\) to \(T^3=S*L^3\).

Now that we have the formula for thickness (really, thickness cubed), we can start solving for the ratio of the thickness of A to the thickness of B. Start with the thickness of diving board B which is: \(T^3=S*L^3\).

Given that diving board A is twice as long and 8 times as stiff as diving board B, we know that the formula for A is: \(T^3=8S*(2L)^3\).

Next divide the thickness (cubed) of diving board A by the thickness (cubed) of diving board B: \(\frac{8S*(2L)^3}{S*L^3}\) = \(\frac{8S*8L^3}{S*L^3}\) = \(\frac{64SL^3}{SL^3}\) = \(64\)

This tells us that the ratio of diving board A's thickness (cubed) to diving board B's thickness (cubed). The final step is to take the cube root of 64, which is 4. The answer.

#Collected from gmatclub forum.

Can you please explain how those two combine?

\(S=T^3\) and \(S=1/L^3\) give:

\(T^3 = 1/L^3\)

\(T^3 * L^3 = 1\)

I understand why we are ignoring the proportionality constant, but I'm unsure how we are combining both to get: \(S=\frac{T^3}{L^3}\)

Correct answer - 4

\(S(A)=\frac{t(A)^3}{{kl(A)^3}} \)

\( l(A)=2l(B) \)

\( S(A)=\frac{t(A)^3}{{k(2l(B)^3}}=\frac{t(A)^3}{{k8l(b)^3}} \)

\( S(B)=\frac{t(B)^3}{{kl(B)^3}} \)

\( S(A)=8S(B)\)

\(\frac{S(A)}{S(B)}=8\)

\(\frac{t(A)^3}{k8l(b)^3}/\frac{t(B)^3}{kl(B)^3}=8\)

\(\frac{{t(A)^3}}{{8t(B)^3}}=8\)

\(\frac{{t(A)^3}}{{t(B)^3}}=64\)

\(\frac{{t(A)}}{{t(B)}}=4\)