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06 Dec 2018, 05:28
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52% (02:11) correct
48%
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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.)
A. 2
B. 4
C. 8
D. 16
E. 64
A. 2
B. 4
C. 8
D. 16
E. 64
Archit3110
Major Poster
06 Dec 2018, 07:03
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Bunuel
stiffness = thickness^3/ length ^3
given length of A= 2 * length of B
and stiffness of A = 8 * stiffness of B
t hickness^3= stiffness* length ^3
find for both a and b
A thickness^3= stiffness* length ^3/ B thickness^3= stiffness* length ^3
or say thickness (A/B) ^3= 8 SB* 2^3* Lb/ sb* lb
so thickness (A/B)^3= 64
or thickness ( A/B) = 4 IMO B
02 Feb 2019, 12:39
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I found Archit3110's answer very difficult to comprehend so here's another way to think about this problem that might help:
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.
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.
