An airplane pilot over the Pacific sights an atoll at the angle of depression of 5°. At this time, the horizontal distance from the airplane to the atoll is 4,535 meters. What is the height of the plane to the nearest meter? *
1 point
395 meters
397 meters
4,518 meters
51,835 meters
To find the height of a pole, a surveyor moves 150 feet away from the base of the pole and then, with a transit 4 feet tall, measures the angle of elevation to the top of the pole to be 42°. To the nearest foot, what is the height of the pole? *
1 point
135 ft
139 ft
108 ft
112 ft
A piece of art is in the shape of an equilateral triangle with sides of 15 in. What is the area of the piece of art to the nearest tenth? *
1 point
73.45 inches squared
84.5 inches squared
97.4 inches squared
169 inches squared
In ∆DEF, m∠D=42°, m∠E=63°, and EF=24 in. What is DE to the nearest tenth of an inch? *Hint: Law of Sines *
1 point
32.0 in
26.0 in
34.6 in
16.6 in
To solve each of these questions, we will use different mathematical concepts and formulas. I will guide you through the steps to find the answer to each question.
1. An airplane pilot over the Pacific sights an atoll at the angle of depression of 5°. At this time, the horizontal distance from the airplane to the atoll is 4,535 meters. What is the height of the plane to the nearest meter?
To solve this problem, we can use trigonometry. The tangent function is useful for finding the height when given the angle of depression and the horizontal distance.
Let h be the height of the plane. We have the following information:
Angle of depression = 5°
Horizontal distance = 4,535 meters
Using the tangent function, we can set up the following equation:
tan(5°) = h / 4,535
To find h, we rearrange the equation:
h = tan(5°) * 4,535
Calculating this using a calculator, we find that the height of the plane is approximately 397 meters. Therefore, the answer is B) 397 meters.
2. To find the height of a pole, a surveyor moves 150 feet away from the base of the pole and then, with a transit 4 feet tall, measures the angle of elevation to the top of the pole to be 42°. To the nearest foot, what is the height of the pole?
In this question, we are given the angle of elevation, the distance from the base of the pole, and the height of the transit. We can use the tangent function again to solve for the height of the pole.
Let h be the height of the pole. We have the following information:
Angle of elevation = 42°
Distance from the base = 150 feet
Height of the transit = 4 feet
Using the tangent function, we can set up the equation:
tan(42°) = (h + 4) / 150
To find h, we rearrange the equation:
h = tan(42°) * 150 - 4
Calculating this using a calculator, we find that the height of the pole is approximately 108 feet. Therefore, the answer is C) 108 ft.
3. A piece of art is in the shape of an equilateral triangle with sides of 15 inches. What is the area of the piece of art to the nearest tenth?
To find the area of an equilateral triangle, we can use the formula:
Area = (sqrt(3) / 4) * (side length)^2
In this case, the side length is given as 15 inches.
Calculating this using a calculator, we find that the area of the piece of art is approximately 97.4 inches squared. Therefore, the answer is C) 97.4 inches squared.
4. In ∆DEF, m∠D = 42°, m∠E = 63°, and EF = 24 in. What is DE to the nearest tenth of an inch? (Hint: Law of Sines)
To solve this triangle, we can use the Law of Sines, which states that the ratio of the sine of an angle to the length of the opposite side is the same for all angles in a triangle.
Let DE be x. We have the following information:
∠D = 42°
∠E = 63°
EF = 24 in
Using the Law of Sines, we can set up the equation:
sin(∠D) / DE = sin(∠E) / EF
Substituting the given values, we have:
sin(42°) / x = sin(63°) / 24
To find x, we rearrange the equation:
x = (24 * sin(42°)) / sin(63°)
Calculating this using a calculator, we find that DE is approximately 16.6 inches. Therefore, the answer is D) 16.6 in.
In all cases, draw a diagram. Looks like you need to review your basic trig functions.
#1. h/4535 = tan5°
#2. (h-4)/150 = tan42°
#3. for a triangle with side s, A = √3/4 s^2
#4. use the hint! F = 180-42-63 = 75°
DE/sin75° = 24/sin42°