Answer:
The height is 3791 ft.
Step-by-step explanation:
You need to make a drawing. Start with a vertical segment, 2 inches long, on the left side of the page. Label the top point A and the bottom point B. At the bottom endpoint, draw a longer horizontal segment, 4 inches long, to the right. Label the bottom right endpoint C. Connect A and C with a segment. Angle C is the original angle of elevation. On the bottom horizontal side, approximately 1 inch to the left of C, draw point D. Connect D and A with a segment. DA = 1000 ft. m<C = 34.8°. m<ABD = 40.4°. We are looking for AB, the height of the volcano.
We can work on triangle ADC.
m<C = 34.8°
Angles ADB and ADC are a linear pair, so m<ADC = 180° - 40.4° = 139.6°
m<DAC + m<ADC + m<C = 180°
m<DAC + 139.6° + 34.8° = 180°
m<DAC = 5.6°
Using the law of sines, we can find AC.
[tex] \dfrac{\sin A}{a} = \dfrac{\sin B}{b} [/tex]
[tex]\dfrac{\sin 5.6^\circ}{1000} = \dfrac{\sin 139.6^\circ}{AC}[/tex]
[tex]AC = \dfrac{1000\sin 139.6^\circ}{\sin 5.6^\circ}[/tex]
[tex] AC = 6642~ft [/tex]
Now we use triangle ABC. AC = hypotenuse. AB = opposite leg. <C is known angle.
[tex] \sin C = \dfrac{opp}{hyp} [/tex]
[tex]\sin 34.8^\circ = \dfrac{AB}{6642~ft}[/tex]
[tex]AB = 6642~ft \times \sin 34.8^\circ[/tex]
[tex] AB = 3791~ft [/tex]
Answer: The height is 3791 ft.
