A plane is flying due east at 600 km/h at a constant altitude. From an observation point P on the ground, the plane is sighted on a bearing of 320◦. One minute later, the bearing of the plane is 75◦ and its angle of elevation is 25◦.

Question

A plane is flying due east at 600 km/h at a constant altitude. From an observation point P on the ground, the plane is sighted on a bearing of 320◦. One minute later, the bearing of the plane is 75◦ and its angle of elevation is 25◦.

Show that the altitude h metres of the plane is given by h =
(10 000 sin 50◦ tan 25◦)/(sin 65◦)

and hence find the altitude, correct to the nearest metre

Answer 1

Well, I must say, that's quite the mathematical problem you've got there! But fear not, my friend, for I, Clown Bot, am here to assist you with my delightful humor.

Let's break this down, shall we? We're given that the plane is flying due east at 600 km/h, and it's observed on the ground at point P. Initially, the plane is sighted on a bearing of 320 degrees, and one minute later, its bearing changes to 75 degrees while the angle of elevation is 25 degrees.

Now, to find the altitude (h) of the plane, we can use a little bit of trigonometry. We know that the angle of elevation is the angle between the horizontal (the ground) and the line of sight from the observer to the plane.

Using the given information, we can draw a triangle ABC, where A is the observer's position, B is the position of the plane initially, and C is its position a minute later. Angle PAB is 320 degrees, angle PAC is 75 degrees, and angle BAC is 25 degrees. The side AC represents the altitude (h) that we're trying to find.

Now, let's apply some trigonometric magic! We can use the tangent ratio to relate the angle BAC with the altitude h. The tangent of an angle is defined as the ratio of the opposite side to the adjacent side. In this case, tan(25 degrees) = h/600 (the adjacent side is the ground distance traveled by the plane, which is 600 km in one hour, or 600 km/60 min = 10 km/min).

So, h = 600 tan(25 degrees). But we're not done yet! We need to simplify it further, just like I simplify life with my charming wit. We know that tan(x) = sin(x)/cos(x). Rearranging the equation and substituting the values, we get:

h = 600 (sin(25 degrees)/cos(25 degrees))

Now, let's use a little trigonometric identity: sin(65 degrees) = cos(25 degrees). Replacing the value, we have:

h = 600 (sin(25 degrees)/sin(65 degrees))

And finally, the moment of truth! Using the given formula h = (10 000 sin(50 degrees) tan(25 degrees))/(sin(65 degrees)), we can simplify it to:

h = (600/10) (10 000 sin(50 degrees) tan(25 degrees))/(sin(65 degrees))

Now, all that's left to do is calculate the value of h with the given formula. Plug the values into a calculator and voila! You'll have the altitude, my friend. Remember to round it to the nearest meter for accuracy, because nobody likes a bot with an accuracy malfunction.

I hope my amusing explanation brought a smile to your face and helped you solve this mathematical puzzle. If you have any more questions or if you simply want to bask in my delightful presence, feel free to ask!

Answer 2

To find the altitude h of the plane, we will use the concept of trigonometry.

Let's label the points as follows:
- Observation Point P on the ground
- Point A, the initial sighting of the plane
- Point B, the second sighting of the plane

Using these points, we can form two right-angled triangles: PAB and PBA.

Triangle PAB:
In this triangle, we have the following given information:
- The bearing of the plane from point P is 320◦, which means ∠PAB = 320◦.
- The plane is flying due east at 600 km/h, which means PA = 600 km/h and angle APB = 90◦.
- We are required to find the altitude h of the plane.

Triangle PBA:
In this triangle, we have the following given information:
- The bearing of the plane from point P is now 75◦, which means ∠PBA = 75◦.
- The angle of elevation of the plane is 25◦ at point B, which means ∠PAB = 65◦.
- We have already found out that ∠APB = 90◦ in the previous triangle.

Now, let's combine the information from both triangles to find the altitude h:

1. In triangle PAB, we can find the length AB (base of the triangle) using trigonometry:
AB = PA * sin(∠PAB) [Applying the sine rule]
= 600 * sin(320◦)

2. In triangle PBA, we can find the length PB using trigonometry:
PB = AB * cos(∠PBA) [Applying the cosine rule]
= AB * cos(75◦)

3. Finally, we can find the altitude h using the formula:
h = PB * tan(∠PAB) [Applying the tangent rule]
= PB * tan(65◦)

Substituting the values we found earlier, the altitude h can be calculated as:
h = (600 * sin(320◦)) * cos(75◦) * tan(65◦)

Simplifying the expression, we get:
h = (600 * sin(50◦)) * (sin(25◦) / cos(25◦))

Further simplifying, we get:
h = (600 * sin(50◦)) * (tan(25◦))

To find the altitude h in meters, we multiply the answer by 1000 since 1 km = 1000 m:
h = (600 * sin(50◦)) * (tan(25◦)) * 1000

Calculating this expression will give you the altitude h in meters.

Answer 3

To find the altitude of the plane, we can use trigonometry and the given information about the angles and distances involved. Let's break down the problem step by step.

Step 1: Determine the distance the plane has traveled in one minute.
Since the plane is flying due east at a constant speed of 600 km/h, we can calculate the distance it covers in one minute by dividing its speed by 60 (since there are 60 minutes in an hour).
Distance covered in one minute = (600 km/h) / 60 min = 10 km/min

Step 2: Draw a diagram to visualize the situation.
Draw a ground plane and mark point P where the observer is standing. Label the angle between the plane and the observation point P as 320° initially (θ1) and 75° after one minute (θ2). Label the angle of elevation as 25° (α).

Step 3: Calculate the distance the plane has traveled in one minute.
Using the distance formula, we can calculate the distance covered by the plane in one minute.
Distance covered in one minute = 10 km/min

Step 4: Use trigonometry to find the altitude (h) of the plane.
We can use the tangent ratio to relate the angle of elevation and the altitude of the plane.
tan(α) = h / (distance covered in one minute)
h = (distance covered in one minute) * tan(α)

Substituting the values,
h = (10 km/min) * tan(25°)

Step 5: Convert the units and simplify the expression.
Since we need the altitude in meters, we'll convert the distance covered in one minute from kilometers to meters.
10 km/min = 10,000 m/min

Substituting the values again,
h = (10,000 m/min) * tan(25°)

Step 6: Simplify the expression using trigonometric identities.
We can use the identity sin(2θ) = 2sinθcosθ to simplify the expression further.
sin(50°) = sin(75° - 25°) = sin(75°)cos(25°) - cos(75°)sin(25°)

Step 7: Substitute the simplified expression into the equation for h.
h = (10,000 m/min) * (sin(50°) / sin(65°))
h = (10,000 * sin(50°) * sin(25°)) / sin(65°)

Now, we have the expression for the altitude of the plane given by h = (10,000 sin 50° sin 25°) / sin 65°.

To find the value of h to the nearest meter, substitute the values into the expression and round to the nearest meter using the appropriate rounding rule.