Question

while at a friends house, michella noticed that he had a window shaped like a right triangle. she realized that she wanted one just like it and decided to calculate its area to figure out how much space it would take up on her wall. when she asked her friend about the dimensions of the window, he told her that the hypotenuse was \\(2.75\text{ m}\\) and the base was \\(2.45\text{ m}\\), but he couldnt remember the height.

use what you know about trigonometry to find the height of the window and calculate the area. round your answers to the nearest tenth.

Explanation:

Identificar datos y fórmulas

Usando el concepto de Resolución de triángulos rectángulos:

• Hipotenusa \(c = 2.75\text{ m}\)
• Base \(b = 2.45\text{ m}\)
• Altura \(h\) es el cateto opuesto al ángulo de elevación \(\theta\).
• Relación trigonométrica: \(\cos(\theta) = \frac{b}{c}\)
• Altura: \(h = c \cdot \sin(\theta)\)

Calcular el ángulo de elevación

Usando la razón del coseno:

$$ \cos(\theta) = \frac{2.45}{2.75} \approx 0.8909 $$

$$ \theta = \arccos(0.8909) \approx 27.01^\circ $$

Calcular la altura de la ventana

Usando la razón del seno:

$$ h = 2.75 \cdot \sin(27.01^\circ) \approx 1.249\text{ m} $$

Redondeando al décimo más cercano:

$$ h \approx 1.2\text{ m} $$

Calcular el área de la ventana

Usando el concepto de Área de un triángulo:

$$ A = \frac{\text{base} \cdot \text{altura}}{2} = \frac{2.45 \cdot 1.249}{2} \approx 1.53\text{ m}^2 $$

Redondeando al décimo más cercano:

$$ A \approx 1.5\text{ m}^2 $$

Answer:

• Altura de la ventana: \(1.2\text{ m}\)
• Área de la ventana: \(1.5\text{ m}^2\)