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Assuming volume, V, is constant, the length of an elliptical solid being sheared varies directly as the square of the radius and inversely as the width. If a solid with a width of 10 cm and radius of 10 cm has a length of 2 cm, what is the length of a solid with a radius of 15 cm and width of 18 cm?
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2 Answers

The problem generates an equality which is:
(L2/L1)=(r22/r12)*(w1/w2)
Here
r2 = 15 cm; r1 = 10 cm; w2=18 cm;w1= 10 cm; L1=2 cm
Plugging this values in the formula, we obtain:
L2=(225/180)*2 = 2.5 cm
Having V as a constraint or boundary conditions all the parameters will be governed by the given proportions.
 
My hint to you is to directly translate expressions using definitions.
 
To say "x varies directly with y" means x is proportional to y i.e. x=ky (or y=kx except written this way, the value of this "k" will be the reciprocal of the other k) for some constant k (where x and y are varying quantities); they are "directly" varying in that if the absolute value of one variable increases, so will that of the other
 
To say "x varies inversely with y" means x is reciprocally related to y i.e. x=k/y (y=k/x) for some constant k (x and y varying); they are "inversely" related in that if the absolute value of one variable increases, the absolute value of the other will decrease.
 
Often, simple problems about variation will first give you enough information to solve for the constant (fixed for ALL combos of x and y or whatever variables are of concern)and require that you use that information to then find the value of one of the variables for a provided value of the other variable.

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