Answer:
711 ft (nearest foot)
Step-by-step explanation:
Create two equations using trig ratios and the information given (refer to the attached diagram). Equate the equations and solve.
First equation
Let y = horizontal distance between sensor 1 and the aircraft
Let h = height of aircraft above the ground
Using the tangent trig ratio:
[tex]\mathsf{\tan(\theta)=\dfrac{opposite \ side}{adjacent \ side}}[/tex]
Given:
- angle = 20°
- side opposite the angle = h
- side adjacent the angle = y
[tex]\implies \mathsf{\tan(20)=\dfrac{h}{y}}[/tex]
Rearrange to make y the subject:
[tex]\implies \mathsf{y=\dfrac{h}{\tan(20)}}[/tex]
Second equation
Let y + 700 = horizontal distance between sensor 2 and the aircraft
Let h = height of aircraft above the ground
Using the tangent trig ratio:
[tex]\mathsf{\tan(\theta)=\dfrac{opposite \ side}{adjacent \ side}}[/tex]
Given:
- angle = 15°
- side opposite the angle = h
- side adjacent the angle = y + 700
[tex]\implies \mathsf{\tan(15)=\dfrac{h}{y+700}}[/tex]
Rearrange to make y the subject:
[tex]\implies \mathsf{y=\dfrac{h}{\tan(15)}-700}[/tex]
Now equate the 2 equations and solve for h:
[tex]\implies \mathsf{\dfrac{h}{\tan(20)}=\dfrac{h}{\tan(15)}-700}}[/tex]
[tex]\implies \mathsf{700=\dfrac{h}{\tan(15)}-\dfrac{h}{\tan(20)}}[/tex]
[tex]\implies \mathsf{700=\dfrac{h\tan(20)-h\tan(15)}{\tan(15)\tan(20)}}[/tex]
[tex]\implies \mathsf{700=\dfrac{\tan(20)-\tan(15)}{\tan(15)\tan(20)}h}[/tex]
[tex]\implies \mathsf{h=\dfrac{700\tan(15)\tan(20)}{\tan(20)-\tan(15)}}[/tex]
[tex]\implies \mathsf{h=710.9678247...}[/tex]
Therefore, the height of the aircraft is 711 ft (nearest foot)
