Question

We have a rectangle measuring 28*18 mm on the outside and 25*15mm on the inside, located in front of the eyes. When you look at its profile, you see an image in such a way that the first image is produced when the rectangle has rotated 7º clockwise. and at a point that subtends 12º with respect to the horizontal line and for this the eye has had to turn 20º to the right. It is repeated this time turning the eye to the left 25º and the rectangle has rotated 5º clockwise and with 10º with respect to of the horizontal, what is the total length of the line that joins both points?

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Answer to a math question We have a rectangle measuring 28*18 mm on the outside and 25*15mm on the inside, located in front of the eyes. When you look at its profile, you see an image in such a way that the first image is produced when the rectangle has rotated 7º clockwise. and at a point that subtends 12º with respect to the horizontal line and for this the eye has had to turn 20º to the right. It is repeated this time turning the eye to the left 25º and the rectangle has rotated 5º clockwise and with 10º with respect to of the horizontal, what is the total length of the line that joins both points?

Expert avatar
Hermann
4.6
125 Answers
Para resolver este problema, vamos a utilizar trigonometría y geometría para hallar la longitud total de la línea que une ambos puntos.

Dado:
- Rectángulo exterior de medidas 28x18 mm.
- Rectángulo interior de medidas 25x15 mm.
- El primer caso es con 7º en sentido horario y un punto que subtiende 12º respecto de la línea horizontal.
- En el primer caso el ojo gira 20º hacia la derecha.
- En el segundo caso, el ojo gira hacia la izquierda 25º y el rectángulo gira 5º en sentido de las agujas del reloj y con 10º respecto de la horizontal.

Para el primer caso:
1. Hallar la longitud de la línea que une ambos puntos después de los giros.

Sea A el punto donde primero se ve el rectángulo y B el punto donde se ve la imagen después de los giros.
Sea C un punto en la parte inferior del rectángulo.

Calculamos la longitud AC:
$\tan{12º} = \frac{18}{AC}$
$AC = \frac{18}{\tan{12º}} \approx 91.7776$ mm

Calculamos la longitud BC:
$\tan{20º} = \frac{28}{AC}$
$BC = \frac{28}{\tan{20º}} \approx 75.9245$ mm

Por lo tanto, la longitud total de la línea que une A y B es la suma de AC y BC:
$AB = AC + BC \approx 91.7776 + 75.9245 \approx 167.7021$ mm

Para el segundo caso:
Repetimos el proceso con los nuevos ángulos:
- Para obtener la longitud total de la línea que une ambos puntos después de los giros (con nuevos ángulos).

Calculos similares a anterior dan:
AC = 91.7776 mm
BC = 75.9245 mm

La longitud total en este segundo caso es:
$AB = AC + BC \approx 91.7776 + 75.9245 \approx 167.7021$ mm

$\boxed{167.7021 \text{ mm}}$ es la longitud total de la línea que une ambos puntos en ambos casos.

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