8. NACA airfoils (from the National Advisory Committee for Aeronautics, predecessor of NASA) are parametrised by two functions f(x) and f(x) representing the upper and lower surfaces of an airfoil cross-section, respectively. These functions can be expressed as fox(x)=tmp(x) [0,1], where > 0 is a vertical scaling parameter, a is a vector of constant parameters, and (x) is a vector collecting a number of pre-defined functions of x: --(0)- 9(x)-> For example, the so-called no012 symmetric airfoil assumes T=0.12 (maximum thickness), a (1.4845,-0.6300-1.7580.1.4215-0.5180), and a =-ay? 0.1 0.05 -0.05 02 0.4 0.6 0.8 fu(x) 3= (a) Show that for symmetric airfoils (ay=a.a =-a) to have zero thickness at x=1, it is necessary that the sum of components of a is zero. (b) Calculate the area of an arbitrary symmetric airfoil cross-section with a = a, a, -a, and a=(aa). Use the result to calculate the cross-sectional area of the n0012 airfoil with the parameters given above. (c) What volume of plastic is required to 3D print ten n0012 airfoil ribs of thickness 0.2, assuming no loss and units of centimetres? (d) Extra (not required for full marks): how would you need to change the functions fu(x) so that the length of the cross-section is not 1, but 10, without changing the shape?