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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