A statue 5m high is standing on a base 8m high. Is an observer's eye is 2.5 m above the ground, how far should he stand from the base of the statue in order that the angle between his lines of sight to the top and bottom of the statue be a maximum.

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To find the distance at which the observer should stand from the base of the statue to maximize the angle between his lines of sight to the top and bottom of the statue, we can use some basic trigonometry.

Let x denote the distance at which the observer should stand from the base of the statue. Then, the horizontal distance from the base of the statue to the observer's eye level is x, and the vertical distance from the base to the observer's eye level is 2.5m.

Using similar triangles, we can set up the following equation:

(5 + 2.5) / x = 5 / (8 + 2.5)

Here, (5 + 2.5) represents the total height of the statue (including the base) and 5 represents the height from the observer's eye level to the top of the statue. Similarly, (8 + 2.5) represents the total height from the base to the top of the statue.

Let's solve this equation step-by-step:

1. Simplify the equation:
7.5 / x = 5 / 10.5

2. Cross-multiply:
7.5 * 10.5 = 5 * x

3. Simplify:
78.75 = 5x

4. Divide both sides by 5:
x = 78.75 / 5
x = 15.75

Therefore, the observer should stand approximately 15.75 meters away from the base of the statue in order to maximize the angle between his lines of sight to the top and bottom of the statue.

To find the distance at which the angle between the lines of sight to the top and bottom of the statue is a maximum, we can use a little bit of trigonometry.

Let's label the height of the statue as h1 (5m), the height of the base as h2 (8m), the height of the observer's eye as h3 (2.5m), and the distance we want to find as x.

To visualize the problem, let's draw a diagram:

*
/ |
h2 + h1 / | h3
/ | |
/ | |
_____________|______|__________________
x

From the diagram, we can see that the angle between the lines of sight to the top and bottom of the statue is the same as the angle between the lines connecting the observer's eye to the top and bottom of the statue.

Let's say that angle is theta.

To find the distance x, we can use the trigonometric tangent function, which is defined as the ratio of the opposite side to the adjacent side of a right triangle.

Using the tangent function, we have:

tan(theta) = h1 / x (opposite / adjacent)

Since we want to maximize the angle theta, we need to find the maximum value of tangent. The maximum value of tangent occurs when the angle is 45 degrees (π/4 radians).

Now, let's solve for x using the tangent function:

tan(theta) = h1 / x
tan(π/4) = h1 / x (substituting the maximum angle)
1 = h1 / x
x = h1

Therefore, the observer should stand a distance of 5 meters from the base of the statue in order for the angle between the lines of sight to the top and bottom of the statue to be a maximum.