Rachel and Mark are on a roadtrip. Just when they think they can\'t take being i

Question

Rachel and Mark are on a roadtrip. Just when they think they can't take being in the car with each other any longer, they see a billboard advertising
Have your picture taken with the buffalo
at Cory & June's
Hexaflexagon Mega-Warehouse
The sign is 15 feet wide, perpendicular to a straight road, and 20 feet from the road. Rachel decides to snap a picture of the sign to take back to show everyone in calculus class. At what point on the road would Rachel have the best view of the billboard as they drive by? That is, at what point on the road is the angle subtended by the billboard a maximum? Is the pair tempted to take the 30-mile detour for the photo-op?

Explanation / Answer

Let x be the distance along the road from a point level with the sign. The viewing angles are given by:

tan = 20/x
tan = (35)/x

where is the viewing angle to the edge of the sign closest to the road and is the angle to the edge farthest from the road. The subtended angle = -

tan = tan( - ) = (tan - tan)/(1 + tantan) = [(35)/x - 20/x]/[1 + (20/x)*(35)/x]
tan =   15x/(x² + 700)

= atan[15x/(x²+700)]

To maximize , set the derivative of with respect to x to zero, solve for x, and evaluate at that value of x where d = d atan(u) = du/(1+u²) and u = 15x/(x² + 700)

d/dx = 0 = [du/dx]/(1+u²) => du/dx = 0
du/dx = [15(x² + 700) - 30x²]/(x²+700)² = 0
10500 - 15x² = 0
x² = 700

x = ±700
x = ±26.46

The subtended angle is a maximum at a point 26.46 feet either side of the sign.
The subtended angle is atan[15x/(x²+700)] = atan(15*26.46/1400) = atan0.2835 = 15.8degree

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