Solve & find Radius of curvature(r) of x^3+y^3=2a^3 & check if r=a/((2)^(1/2))?

Question

Pls don't jus tgive me the answer, but just give the solution showing the steps involved in solving this problem

Answer 1

I'm afraid you're going to have to get your hands dirty and calculate.

Note that there are two approaches you can take to start the calculations. First, you can express y explicitly as a function of x and start differentiating. Second, you can just note that the derivate of the LHS = 0, sort of in the style of implicit differentiation.

Answer 2

The radius of curvature is not a constant. Atleast for this form of curve. Better look into the physical sense of it. The radius of curvature of this particular curve is,

r=-(2x(2a^3-x^3)^(-2/3)+(x^5)(2a^3-x^3)^(-5/3))