Banner Outdoor

How far from the base of the CN Tower should she stand?

Protestors unfurl a banner from the outdoor observation deck of the CN Tower, 342m above ground level. The banner is 20m long, and hangs vertically. To get the best picture, a news photographer 2m tall, standing at ground level, would like to stand where the banner appears largest. How far from the base of the CN Tower should she stand? Namely, what distance from the CN Tower maximizes the visual angle of the photographer?

As Dave B says, cute problem.

I *think* 80√17 metres … approximately 329.85 metres.

I did it by Pythagoras and calculus. Call the photographer’s distance from the base “x”.

Sketch it out. Once you subtract the photographer’s height, you’ve got two right-angled triangles on a common base x. The one up to the foot of the banner has height 320. The one up to the top of the banner has height 340.

This means that the triangle formed by the photographer’s eye, the top of the banner and the bottom of the banner has sides 20, √(320^2 + x^2) and √(340^2 + x^2)

Plug those sides into the cosine rule, and you get a function in x for the cosine of the angle subtended by the banner at the photographer’s eye.

f(x) = (2x^2 + 217600)/(2 * √(320^2 + x^2) * √(340^2 + x^2))

The angle is a maximum when the cosine is a minimum, so find the derivative f’(x) and solve for f’(x) = 0

I had to use a math package at that stage, but it gave 80√17. It’s such a clean result that I strongly suspect there might be an easier way.

How to make an outdoor banner from start to finish!

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