An observer on the ground at point A watches a rocket ascend. The observer is 120 feet from the launch point B. As the rocket rises, the distance d from the observer to the rocket increases.
a. Express the measure of angle A in terms of d.
b. Find the measure of angle A if d = 1500 feet.
c. Find the measure of angle A if d = 2000 feet.
a) after you make your sketch you should see that
cos A = 120/d
b) if d=1500
cosA = 120/1500 = .08
A = appr 5.4°
( here is what I did on my calculator .....
120
÷ 1500
=
2ndF
cos
=
to get 85.4114...
c) repeat the steps I showed you in b)
let me know what you got
To solve this question, we can use trigonometry and the concept of right triangles. Let's break down each part of the question:
a. To express the measure of angle A in terms of d, we can use the tangent function since we have the opposite (120 feet) and adjacent sides (d feet) of the right triangle formed by the observer, the rocket, and the launch point. The tangent function is defined as:
tangent(A) = opposite/adjacent
In this case, the opposite side is 120 feet, and the adjacent side is d feet. So, we have:
tangent(A) = 120/d
To express angle A in terms of d, we can take the inverse tangent (also known as arctan) of both sides:
A = arctan(120/d)
b. To find the measure of angle A when d = 1500 feet, we substitute this value into the expression we derived in part (a):
A = arctan(120/1500)
Using a calculator, we can find the value of arctan(120/1500) is approximately 4.642 degrees.
c. Similarly, to find the measure of angle A when d = 2000 feet, we substitute this value into the expression derived in part (a):
A = arctan(120/2000)
Again, using a calculator, we find that arctan(120/2000) is approximately 3.436 degrees.
So, the measure of angle A is approximately 4.642 degrees when d = 1500 feet, and approximately 3.436 degrees when d = 2000 feet.
To answer these questions, we can use trigonometry. Let's start with part (a) and express the measure of angle A in terms of d.
a. Express the measure of angle A in terms of d:
Since we have a right triangle with the observer at point A, the launch point at point B, and the distance d increasing as the rocket rises, we can use the tangent function to relate the angle A to the distance d. The tangent function is defined as the opposite side divided by the adjacent side in a right triangle.
In this case, the opposite side is d (the distance between the observer and the rocket) and the adjacent side is 120 feet (the distance between the observer and the launch point). Therefore, we can express the measure of angle A in terms of d using the tangent function:
tan(A) = opposite side / adjacent side
tan(A) = d / 120
b. Find the measure of angle A if d = 1500 feet:
To find the measure of angle A when d = 1500 feet, we substitute d = 1500 into the equation we found in part (a) and solve for A:
tan(A) = 1500 / 120
Using a scientific or graphing calculator, we can find the inverse tangent function (tan^(-1)) of both sides to find angle A:
A ≈ atan(1500 / 120)
A ≈ 86.09 degrees
Therefore, the measure of angle A is approximately 86.09 degrees when d = 1500 feet.
c. Find the measure of angle A if d = 2000 feet:
Similarly, to find the measure of angle A when d = 2000 feet, we substitute d = 2000 into the equation we found in part (a) and solve for A:
tan(A) = 2000 / 120
Using the inverse tangent function, we can find angle A:
A ≈ atan(2000 / 120)
A ≈ 87.64 degrees
Therefore, the measure of angle A is approximately 87.64 degrees when d = 2000 feet.