Question

Статуя высотой 4 м стоит на колонне, высота которой 5,6 м. На каком расстоянии должен встать человек ростом (до уровня глаз) 1,6 м, чтобы видеть статую под наибольшим углом? 

Answer 1

L-расстояние до статуи
Угол, под которым человек видит подножие статуи tg α=(5,6-1,6)/L =4/L
Угол, под которым человек видит вершину статуи tg β=(4+5,6-1,6)/L=8/L
Угол, под которым человек видит статую ф=β-α
tgф=(tgβ-tgα)/(1+tgα*tgβ)=4L/(L²+32)
Продифференцируем по L и приравняем к нулю:
4(L²+32)-8L²=0
L²=32
L=4√2~2, 8 м

Answer 2

Calculating the Distance to See the Statue at the Greatest Angle

To calculate the distance at which a person should stand to see the statue at the greatest angle, we can use similar triangles.

Let's denote the height of the statue as h (4 meters) and the height of the column as H (5.6 meters). The person's height (up to eye level) is P (1.6 meters). We need to find the distance at which the person should stand, which we'll call D.

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

h / H = P / (H + D)

To solve for D, we can cross-multiply and rearrange the equation:

h * (H + D) = P * H

Expanding the equation:

h * H + h * D = P * H

Simplifying:

D = (P * H - h * H) / h

Now we can substitute the given values into the equation:

D = (1.6 * 5.6 - 4 * 5.6) / 4

Calculating the result:

D = (8.96 - 22.4) / 4

D = -13.44 / 4

D = -3.36

The negative value indicates that the person should stand on the opposite side of the statue from the column. However, in this case, it doesn't make sense to have a negative distance. Therefore, we can take the absolute value of D to get the positive distance:

D = 3.36 meters

So, the person should stand approximately 3.36 meters away from the statue to see it at the greatest angle.

Please note that this calculation assumes that the person is standing on level ground and that the statue is positioned at the top of the column.