A child’s toy is a flat disk shaped as in the diagram, a square with semicircular ends. The side of the square is measured tobe 8 cm with an error of 0.05 cm

Question

A child’s toy is a flat disk shaped as in the diagram, a square with semi‐circular ends. The side of the square is measured to be 8 cm with an error of 0.05 cm. Use differentials to find the percent error in the area. http://imgur.com/dT08Kv4

Answer 1

So the area is area of square + Area of semicircles.

Let X be length of one side of square

area of square is: x^2

area of one semicircle can be found by the fact that the radius is (X/2)

so area of semicircle is (1/2)*pi*(X/2)^2 = (1/2)*pi*x^2/4

 

total area : area of square + 2times area of semicircle

A = x^2 + pi*x^2/4 = x^2(1+pi/4). Plug in 8cm for X

 

take derivative wrt x

dA/dx = 2x(1+pi/4)

so dA = 2x(1+pi/4)dx

 

plug in values X = 8 cm and dx = .05 cm

 

dA/A = 2dx/X = 2(.05)/8 = .1/8 = .0125 = 12.5 %