Problem 24 - Algebra 2

Question

An observer on the ground at point \(A\) watches a rocket ascend. The observer is 1200 ft from the launch point \(B\) . As the rocket rises, the distance \(d\) from the observer to the rocket increases. a. Write a model for \(m \angle A .\) b. Find \(m \angle A\) if \(d=1500\) ft. Round your answer to the nearest degree. c. Find \(m \angle A\) if \(d=2000\) ft. Round your answer to the nearest degree.

Step-by-Step Solution

Answer

The model for the measure of angle A as the rocket ascends is given by m ∠A = arctan(1200/d). For a distance d of 1500ft, m ∠A is approximately 39 degrees. For a distance d of 2000ft, m ∠A is approximately 31 degrees.

Step 1: Understanding the Problem

The situation can be modelled as a right triangle where the observer at point A, the launch point B, and the rocket form the triangle. At any moment, the distance from A to B (ground distance) is a constant (1200 ft) and we can say it is the adjacent side. The distance from the observer to the rocket (d) is the hypotenuse. We're asked to find the measure of the angle A (adjacent to the hypotenuse), which depends on the changing value of d.

Step 2: Write the Model

The tangent function (tan) gives the ratio of the opposite side to the adjacent side in a right triangle. Since m ∠A changes depending on d, we can use tan to find the measure of the angle: tan(m ∠A) = AB/d (adjacent/hypotenuse). Solving for m ∠A gives m ∠A = arctan(AB/d).

Step 3: Find m ∠A for d = 1500ft

Substitute d = 1500 into the model: m ∠A = arctan(1200/1500). Computing this gives m ∠A approximately to 38.66 degrees. The nearest degree would be 39 degrees when rounded off.

Step 4: Find m ∠A for d = 2000ft

Now substitute d = 2000 into the model: m ∠A = arctan(1200/2000). Computing this gives m ∠A approximately to 30.96 degrees. The nearest degree would be 31 degrees when rounded off.

Key Concepts

Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees.
This angle is called the "right angle." The right triangle is fundamental in trigonometry.
The sides of a right triangle have specific names:

• Hypotenuse: The longest side of a right triangle, opposite the right angle.
• Adjacent side: The side next to the angle of interest, not the hypotenuse.
• Opposite side: The side across from the angle of interest.

In our exercise, the observer, launch point, and rocket form a right triangle.
The distance between A and B is a constant 1200 ft, representing the adjacent side when we focus on angle A.
The distance from the observer to the rocket (d) changes, acting as the hypotenuse.

Tangent Function

The tangent function, often written as "tan," is one of the basic functions in trigonometry.
It is crucial for finding unknown angles and sides in right triangles.
The function is defined as the ratio between the opposite side and the adjacent side of a triangle:

• \( an( heta) = \frac{\text{opposite}}{\text{adjacent}}\)

In our problem, we're trying to find the angle at point A.
Since we know the sides adjacent and opposite to this angle, the tangent function becomes useful.
Given that the rocket's height is not known, using the tangent function can model angle A as \( an(m \angle A) = \frac{1200}{d}\).
This helps in establishing a relationship between the angle and the continuously changing distance d.

Angle of Elevation

The angle of elevation is the angle formed by the line of sight of the observer looking upwards at an object above the horizontal.
In this scenario, it's the angle at which the observer must tilt their gaze upwards to see the rocket.
As the rocket ascends, this angle changes, and it's crucial in determining its position relative to the observer.

In the right triangle setup, the angle of elevation is the angle at point A.
To find this angle, we use the model established by the tangent function.
The angle's size depends on the variable distance d, so as the rocket gets farther, the angle decreases.
This is evidenced by finding specific angle measurements when different values of d (1500 ft and 2000 ft) are substituted into our function.

Arctangent

The arctangent, denoted as \( ext{arctan}\) or \( an^{-1}\), is the inverse function of the tangent.
It helps find an angle when we know the tangent ratio.
Thus, if \( an(\theta) = x\), then \(\theta = \text{arctan}(x)\).

For our problem, the arctangent is key to determining the angle of elevation ( \(m \angle A\)) once we have the ratio of opposite to adjacent sides.
By calculating \(\text{arctan}\left(\frac{1200}{d}\right)\), we find the angle for different values of d.

• When \(d = 1500\) ft, \(\text{arctan}(0.8)\) gives approximately 39 degrees.
• When \(d = 2000\) ft, \(\text{arctan}(0.6)\) yields approximately 31 degrees.

This process directly applies the inverse function to output an angle measurement.
Understanding arctangent is vital for solving similar problems where angle measurements are required based on side length ratios.