A rectangular billboard must cover an advertising area of 30m2. The material used for the frame along the perimeter costs 25 per meter, while the special vinyl covering the face incurs a cost that grows with width according to c(w)=50e0.05w dollars per square meter, where w is the width (in meters) of the billboard. The height of the billboard is denoted h (in meters).
(a) Write a formula for the total cost C(w) of constructing the billboard in terms of w only.
(b) Compute C′(w) and simplify.
(c) Find all critical widths w>0 that satisfy C′(w)=0.
(d) Use a second--derivative test or sign analysis to justify which critical width minimizes the total cost.
(e) Determine the corresponding height and calculate the minimal total cost to the nearest dollar.