Sketch a diagram describing the given situation, and then recall that time is distance divided by velocity.
m∠ E_2≈ 85^(∘) , t=0.5 min
Practice makes perfect
We are given that an airplane at a constant altitude a flies a horizontal distance d toward us at velocity v, and observing for time t we measure angles of elevation ∠ E_1 and ∠ E_2 at the start and end of our observation. Let's sketch a diagram describing this situation.
Now let's substitute the values we are given in our exercise. Additionally, we will label the horizontal distance between us and the position of the airplane at the end of our observation by x.
To find the value of x and the measure of ∠ E_2, we will use one of the trigonometric ratios. Let's recall that the tangent of an angle is the ratio between the side opposite this angle and the side adjacent to this angle. Using this definition, we can write equations for tan 50^(∘) and tan E_2.
tan 50^(∘)=4/x+3 & (I) tan E_2=4/x & (II)
Now we will solve the system of equations using the Substitution Method. Our first step will be to solve the first equation for x.
The value of x is approximately 0.36. Next we will substitute the value of x into the second equation to find tan E_2. Then we will rewrite the equation using the inverse tangent to find the measure of ∠ E_2.
tan E_2=4/0.36
⇓
E_2=tan^(-1)(4/0.36)≈ 85^(∘)
The measure of ∠ E_2 is approximately 85^(∘). Finally, as we are also asked to evaluate the time, let's recall that time is the quotient of distance and velocity.
t=d/v
To find the time of our observation, we will substitute 3 for d and 6 for v.
t=3/6=0.5
The time of our observation is 0.5 minutes or 30 seconds.
