The bottom of a picture five feet tall is four feet above the level of an observer's eyes. How far from the wall on which the picture hangs should the observer stand in order that the picture subtends the greatest possible angle at the observer's eyes?

Question

The bottom of a picture five feet tall is four feet above the level of an observer's eyes. How far from the wall on which the picture hangs should the observer stand in order that the picture subtends the greatest possible angle at the observer's eyes?

Answer 1

Let

Ø = angle to bottom of picture
θ = angle subtended by the picture
x = distance from wall to observer

tanØ = 4/x
tan(θ+Ø) = 9/x
so
(tanθ+tanØ)/(1-tanθtanØ) = 9/x
(tanθ+(4/x))/(1-(4/x)tanθ) = 9/x

see what you can do with that.

or, if
y=distance from eye to bottom of picture
z= distance to top of picture,

x^2+16 = y^2
x^2+81 = z^2
Now the law of cosines says that

5^2 = y^2+z^2-2yz cosθ
25 = (x^2+16)+(x^2+81)-2√(x^2+16)√(x^2+81) cosθ

Answer 2

To find the distance from the wall on which the picture hangs where the observer should stand in order to maximize the angle at the observer's eyes, we need to consider some geometry and trigonometry.

Let's start by visualizing the problem. Imagine a vertical wall with a picture hanging from it. The observer's eyes are located at a certain height, and we want to determine the distance between the wall and the observer where the angle subtended by the picture at the observer's eyes is maximum.

First, let's consider a right-angled triangle formed by the observer's eyes, the bottom of the picture, and the point on the wall where the picture meets.

The height of the picture is given as five feet, and the bottom of the picture is four feet above the observer's eye level. This means that the distance from the observer's eyes to the bottom of the picture is five - four = one foot.

Now, let's denote the distance from the wall to the observer as "x" (in feet). We can set up a proportion based on the similar triangles formed by the observer's eyes, the bottom of the picture, and the point on the wall where the picture meets:

x / (x + 4) = 1 / 5

This proportion arises from the fact that the ratio of the opposite side (x) to the adjacent side (x + 4) in the smaller triangle is equal to the ratio of the opposite side (1) to the adjacent side (5) in the larger triangle.

To find the value of x, we can cross-multiply and solve for x:

5x = x + 4
4x = 4
x = 1

Therefore, the observer should stand one foot from the wall in order for the picture to subtend the greatest possible angle at the observer's eyes.