A painting is hung on a wall in such a way that its upper and lower edges are 10 ft and 7ft above the floor, respectively. An observer whose eyes are 5 ft above the floor stands x feet from the wall. How far away should the observer stand to maximize the angle subtended by the painting

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To find the distance at which the observer should stand to maximize the angle subtended by the painting, we can use trigonometry.

Let's start by drawing a diagram:

_______________(wall)
| \
| \
| \
| \
| \
| \
|_______\(observer)

Given:
- The upper edge of the painting is 10 ft above the floor.
- The lower edge of the painting is 7 ft above the floor.
- The observer's eyes are 5 ft above the floor.

Let the height of the painting be h = 10 - 7 = 3 ft.

Now, let's consider a right triangle formed by the observer, the wall, and the line connecting the observer's eyes to the lower edge of the painting. The hypotenuse of this triangle represents the distance at which the observer should stand to maximize the angle subtended by the painting.

Using the Pythagorean theorem, the length of the hypotenuse can be found:

hypotenuse^2 = (distance from wall)^2 + (height of painting)^2

Let 'd' represent the distance from the wall, then:

hypotenuse^2 = d^2 + 3^2

Since we want to maximize the angle subtended by the painting, we want to find the maximum value of the angle θ.

By definition, the tangent of an angle θ is equal to the opposite side divided by the adjacent side:

tan(θ) = (height of painting) / (distance from wall)

tan(θ) = 3 / d

To maximize the angle θ, we need to find the maximum value of tan(θ).

Differentiating the equation tan(θ) = 3 / d with respect to d, we get:

dθ/d = (-3) / d^2

Setting the derivative equal to zero (to find a maximum or minimum), we have:

(-3) / d^2 = 0

Solving for d, we find that d = 0 is not a valid solution, so:

d^2 = 0

Taking the square root on both sides, we find:

d = 0

However, since distance cannot be zero in this context, we conclude that there is no maximum value for the angle subtended by the painting. This means that the distance at which the observer should stand is not limited, and the observer can move as far away as desired to obtain a larger angle.