SOLUTION: The stiffness of a beam varies jointly as its breadth and depth and inversely as the square of its length. Find the change in stiffness of each of the three dimensions is increased

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Question 927454: The stiffness of a beam varies jointly as its breadth and depth and inversely as the square of its length. Find the change in stiffness of each of the three dimensions is increased by 10%
Found 2 solutions by Theo, josmiceli:
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
s = k*b*d/l^2
s = stiffness of a beam.
b = breadth
d = depth
l = length
k = constant of variation.

if b and d and l are each increased 10%, then the resulting equation becomes:

s = (k * 1.10*b*1.10*d)/(1.10*l)^2

this becomes:

s = k * 1.10^2*b*d/(1.10^2*l^2)

it appears that the 1.10^2 in the numerator and the 1.10^2 in the denominator cancel each other out and you are left with:

s = k*b*d/l^2

this looks a lot like your original equation.

in other words, if you increase each of the 3 dimensions by 10%, the net change in the stiffness of the beam is 0.

let's give some values to b and d and l and see if that works the way it looks like it's working.

assume:
b = 5
d = 10
l = 15

assume the constant of variation is equal to 45.

you get:

s = 45 * 5 * 10 / 15^2 which becomes:

s = 2250 / 225 which becomes:

s = 10

now we'll increase each of the dimensions by 10%.

5 becomes 5.5
10 becomes 11
15 becomes 16.5

formula becomes:

s = 45 * 5.5 * 11 / 16.5^2 which becomes:

s = 2722.5 / 272.25 which becomes:

s = 10

s remained the same.

looks like the formula works and there is no net change if all of the dimensions are each increased by 10%.

this is because the 1.10^2 in the numerator and the 1.10^2 in the denominator cancel out.

here's a reference that talks about joint inverse combined variation type word problems.

http://www.shelovesmath.com/algebra/beginning-algebra/direct-inverse-and-joint-variation/#JointVariationWordProblem

Answer by josmiceli(19441) (Show Source):
You can put this solution on YOUR website!
Let = stiffness
Let = breadth
Let = depth
Let = length
Let = a constant
----------------------
given:

There is no change in the stiffness
Hope that makes sense