a customer went to a boutique and asks the tailor to design a table cloth in the shape of hypocycloid with the equation r(t)= cos^3 i + sin^3 j . she insists to have laces at every edge of the table cloth. calculate the lenght of laces that the tailor have to buy.

Answer 1

Assuming you mean that the edge of the cloth is to be trimmed with lace, you want the arc length of the graph.

ds^2 = (3cos^2(t)sin(t))^2 + (3sin^2(t)cos(t))^2 dt
= 9sin(t)cos(t)(cos(t)+sin(t)) dt
integrate that from 0 to pi/2 and multiply that by 4 to get the entire length of lace.

Or, there are various formulas for general arc lengths of hypocycloids, but they will need you to use the form

x = 1/4 (3cos(t)+cos(3t))
y = 1/4 (3sin(t)-sin(3t))