Create an image where a medium-sized aircraft is seen flying high in the bright daylight sky at an altitude of 880 m. Down on the ground, a Black woman named Sarah is observing the plane. She's carefully calculating the angle of elevation to the plane, initially noted as 67'40', and 25 seconds later as 24'30'. Around her, there are no visible markings, text, or numbers, just imagery depicting this scenario.

A plane is flying at an altitude of 880 m. Sarah is standing on the ground and she observes the angle of elevation to the plane as 67'40'. Twenty-five seconds later the angle of elevation had changed to 24'30'.

(a) How far has the plane flown in that time?
(b) what is the speed to the nearest Km/h of the plane?

Please HELP
I have tried tan67'40'*880 and then the other angle and i subtract but it didn't work
What do i do

My diagram has Sarah at A

and the first position of the plane at P and the second position at Q
The point on the ground below P is B , and below Q the point is C
triangles APB AND AQC are both right-angled and PB = QC = 880 m

In triangle APB
tan67°40' = 880/AB
AB = 880/tan67°40' ----> I see you multiplied
AB = 361.51 m

in triangle AQC
tan 24°30' = 880/AC
AC = 1930.98 m

distance covered by plane in that time
= (1930.98 - 361.51) m
= 1569.47 m

speed = distance/time
=1569.47 m/25seconds
= 1.56947 km/(25/3600) hrs
= appr 226 km/h

review the cotangent function. Draw a diagram and you will see that the distance the plane has flown is

880 cot24°30' - 880 cot67°40'

Well, it seems like you're having a bit of trouble with this problem. Don't worry, I'm here to help! Let's break it down step by step, using a bit of humor along the way.

(a) First, let's find the difference in the angles of elevation. We have 67'40" - 24'30". Now, it's not the time for "subtractraction", just some good old-fashioned subtraction. Just remember to carry the one if you need to!

(b) After getting the difference in angles, convert it to degrees because planes prefer degrees over minutes and seconds. And let's be honest, degrees are much easier to deal with. Just like choosing between a clown car and a real car – the clown car might be more entertaining, but the real car is more practical.

(c) Now, we can use some trigonometry! The tangent of the angle of elevation gives us the ratio between the distance the plane has flown and its altitude. So, we can set up an equation with the tangent of the angle, the altitude, and the distance flown.

(d) Solving for the distance, we get the equation: distance = altitude / tangent(angle). Remember, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. Think of it as the quirky uncle who always sits beside you at family dinners.

(e) Plug in the altitude and the angle in degrees to find the distance the plane has flown. To make things easier, let's use meters for now, because we're going to convert it to kilometers later, just like clowns changing into their regular clothes after a show.

(f) Now, we know that the plane has flown a certain distance in 25 seconds. So, to find the speed in km/h, we first need to convert the time from seconds to hours. An hour is 60 minutes, just like watching a clown show that lasts for 60 minutes – you never know what surprises will happen during that time!

(g) Divide the distance (in kilometers) by the time (in hours) to get the speed of the plane in km/h. Finally, we have our answer!

So, don't worry, my friend. Just follow these steps, keep a positive attitude, and remember that even the trickiest problems can't resist a good laugh!

To find the distance the plane has flown, you can use trigonometry.

Let's first convert the angles from degrees, minutes, and seconds to decimal degrees for ease of calculation.

67'40' = 67 + 40/60 = 67.67 degrees
24'30' = 24 + 30/60 = 24.5 degrees

Let's assume the distance the plane has flown as "d" meters.

Now we can set up two right-angled triangles, one for each observation:

First observation triangle:
In this triangle, the opposite side is the altitude of the plane (880 m) and the angle of elevation is 67.67 degrees.

Using the trigonometric function tangent (tan):
tan(67.67) = 880 / d

Solving for d:
d = 880 / tan(67.67) ≈ 418.93 meters

Second observation triangle:
In this triangle, the opposite side is still the same altitude of the plane (880 m) but the angle of elevation has changed to 24.5 degrees.

Using the same function tangent (tan):
tan(24.5) = 880 / (d + x) ------------------- (1)

In equation (1), "x" represents the distance the plane has flown in the additional 25 seconds.

Now we can substitute the value of "d" from the first equation into the second equation:

tan(24.5) = 880 / (418.93 + x)

Next, we can solve for "x":

x = (880 / tan(24.5)) - 418.93 ≈ 480.42 meters

(a) Therefore, the plane has flown approximately 480.42 meters in 25 seconds.

To find the speed of the plane, we need to convert the distance into meters per second and then convert it to kilometers per hour.

First, let's convert the time to hours:

25 seconds = 25 / 3600 = 0.00694 hours

Next, the speed can be calculated:

Speed = Distance / Time

Speed = 480.42 meters / 0.00694 hours = 69279.05 meters per hour

Finally, converting meters per hour to kilometers per hour:

Speed = 69279.05 meters per hour ≈ 69.28 kilometers per hour

(b) Therefore, the speed of the plane is approximately 69.28 Km/h.

To solve this problem, you can use trigonometry and the concept of similar triangles.

(a) To find the distance the plane has flown, you need to find the difference in the angles of elevation and relate it to the distance traveled by the plane.

Let's define the first angle of elevation as θ1 = 67°40' and the second angle of elevation as θ2 = 24°30'.

First, convert the angle measurements from degrees, minutes, and seconds to decimal degrees for easier calculations.
θ1 = 67°40' = 67 + (40/60) ≈ 67.67°
θ2 = 24°30' = 24 + (30/60) ≈ 24.5°

Next, let's define the vertical distance between Sarah and the plane as h = 880 m, and the horizontal distance traveled by the plane as x (which is what we need to find).

With this information, we can set up the following triangle:
/|
/ |
h/ | x
/ |
/____|

In this triangle, the vertical side represents the altitude of the plane (880 m). The horizontal side represents the distance traveled by the plane (x), and the angle between them is the change in the angle of elevation (θ1 - θ2).

Using trigonometry, we know that:

tan(θ1 - θ2) = h / x

Substituting the values we have:

tan(67.67° - 24.5°) = 880 / x

Now, we can solve for x:

x = 880 / tan(67.67° - 24.5°)

Calculate the value of tan(θ1 - θ2) using a calculator:
tan(67.67° - 24.5°) = 2.998

Substitute this value back into the equation:

x = 880 / 2.998 ≈ 293.62 m

Therefore, the plane has flown approximately 293.62 meters in that time.

(b) To find the speed of the plane in kilometers per hour, we need to convert the distance traveled to kilometers and then divide it by the time it took for the change in angle of elevation to occur.

Given that 25 seconds have passed, we need to convert this time to hours:

Time = 25 / 3600 (1 hour = 3600 seconds)

Now, let's convert the distance traveled to kilometers:

Distance (in km) = 293.62 m / 1000

Speed = Distance / Time

Substituting the values:

Speed = (293.62 m / 1000) / (25 / 3600)

Speed ≈ 0.424 km/h

Therefore, the speed of the plane, to the nearest kilometer per hour, is approximately 0.424 km/h.