1.
A 14 foot ladder is used to scale a 13 foot wall. At what angle of elevation must the ladder be situated in order to reach the top of the wall?
Correct Answer
A. 68.2 degrees
Explanation
To reach the top of the 13-foot wall with a 14-foot ladder, the ladder must be inclined at an angle that allows it to extend beyond the top of the wall. This angle is known as the angle of elevation. In this case, the ladder is longer than the height of the wall, so the angle of elevation will be greater than 45 degrees. The correct answer of 68.2 degrees suggests that the ladder needs to be inclined at a steep angle to reach the top of the wall.
2.
What are the two special triangles?
Correct Answer(s)
A. 45-45-90
C. 30-60-90
Explanation
The two special triangles mentioned are the 45-45-90 triangle and the 30-60-90 triangle. In a 45-45-90 triangle, the two legs are congruent, and the hypotenuse is √2 times the length of the legs. In a 30-60-90 triangle, the shorter leg is half the length of the hypotenuse, and the longer leg is √3 times the length of the shorter leg. These special triangles have specific angle measurements and side ratios that make them useful in solving geometric problems.
3.
Find the length of the opposite side.
Correct Answer
B. 16.10
Explanation
The length of the opposite side can be found by using the Pythagorean theorem. According to the theorem, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, if 16.10 is the length of the hypotenuse and 13.19 is one of the other sides, we can find the length of the opposite side by taking the square root of the difference between the square of the hypotenuse and the square of the known side.
4.
Find side BC.
Correct Answer
B. 4.67
5.
Find the distance of a boat from a lighthouse if the lighthouse is 100 meters tall, and the angle of depression is 6°.
Correct Answer
C. 952.4
Explanation
The angle of depression is the angle formed between a horizontal line and the line of sight from an observer to a point below the observer. In this question, the angle of depression is given as 6°. To find the distance of the boat from the lighthouse, we can use the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the lighthouse (100 meters) and the adjacent side is the distance we are trying to find. By rearranging the formula, we get the distance = opposite side / tangent(angle). Plugging in the values, the distance is approximately 952.4 meters.
6.
Find the central angle of a circle if the arc length is 16 inches and
the radius is 4 inches.
Correct Answer
B. 4
Explanation
The central angle of a circle can be found using the formula: angle = (arc length / radius). In this case, the arc length is given as 16 inches and the radius is given as 4 inches. Substituting these values into the formula, we get angle = (16 / 4) = 4. Therefore, the central angle of the circle is 4.
7.
What is 120° in radians?
Correct Answer
A. 2/3 π
Explanation
The question is asking for the radian measure of 120°. To convert from degrees to radians, we use the formula π/180. So, 120° is equal to (120 * π) / 180 = 2/3 π.
8.
What is 75° in radians?
Correct Answer
C. 5/12 π
Explanation
To convert degrees to radians, we use the formula: radians = (degrees * π) / 180. In this case, we have 75 degrees, so the conversion would be (75 * π) / 180. Simplifying this expression gives us 5/12 π, which is the answer provided.
9.
What is 40° in radians?
Correct Answer
A. 2/9 π
Explanation
To convert degrees to radians, we use the formula radians = degrees * (π/180). In this case, we have 40°, so the calculation would be 40 * (π/180). Simplifying this expression gives us 2/9 π, which is the correct answer.
10.
What is 4/3 π in degrees?
Correct Answer
A. 240°
Explanation
The given question asks for the value of 4/3 π in degrees. To find this value, we need to convert π radians to degrees. Since there are 180 degrees in π radians, we can multiply 4/3 by 180 to get the result. Evaluating this expression gives us 240 degrees. Therefore, the correct answer is 240°.
11.
What is 7/8 π in degrees?
Correct Answer
B. 157.5 °
Explanation
To convert 7/8 π to degrees, we need to multiply it by 180/π since there are 180 degrees in a circle. Simplifying, we get 7/8 * 180/π = 157.5 degrees. Therefore, the correct answer is 157.5 °.
12.
What is 2/5 π in degrees?
Correct Answer
A. 72°
Explanation
To convert radians to degrees, we use the formula: radians * (180/π). In this case, we have 2/5 * π. Multiplying 2/5 by (180/π) gives us 72°. Therefore, the correct answer is 72°.
13.
A photographer points his camera to the top of a building forming an angle of elevation of 50°. If he stands 70 meters from the building, how tall is the building?
Correct Answer
C. 83.4 meters
Explanation
The photographer is standing 70 meters away from the building and the angle of elevation is 50°. To find the height of the building, we can use the tangent function, which is the opposite side divided by the adjacent side. In this case, the opposite side is the height of the building and the adjacent side is the distance from the photographer to the building. Using the formula tan(50°) = height/70, we can solve for the height and find that it is approximately 83.4 meters.
14.
A man is walking along a straight road. He notices the top of a tower makes an angle of 60^{o} with the ground at the point where he is standing. If the height of the tower is h = 15 m, then what is the distance of the man from the tower?
Correct Answer
A. 8.67 meters
Explanation
The man is standing at the base of the tower and notices that the top of the tower makes an angle of 60 degrees with the ground. This forms a right triangle, with the height of the tower being the opposite side and the distance of the man from the tower being the adjacent side. Using trigonometry, we can use the tangent function to find the distance of the man from the tower. tan(60) = opposite/adjacent, so tan(60) = 15/distance. Solving for distance, we get distance = 15/tan(60) = 8.67 meters.
15.
A tree is 200 ft tall. If a man is standing 75 ft away from the tree,
what is the angle of elevation to the bird he is looking at?
Correct Answer
C. 69.4°
Explanation
The angle of elevation is the angle formed between the line of sight from the observer to the object and the horizontal line. In this case, the man is standing 75 ft away from the tree and looking at a bird on top of the tree. To find the angle of elevation, we can use the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the tree (200 ft) and the adjacent side is the distance from the man to the tree (75 ft). Therefore, the tangent of the angle of elevation is 200/75, which is approximately 2.67. Taking the inverse tangent of 2.67, we find that the angle of elevation is approximately 69.4°.
16.
How tall is the flagpole?
Correct Answer
A. 33.32
17.
Find the height.
Correct Answer
C. 173.21
Explanation
The answer 173.21 is the height because it is the largest value given among the numbers 30.48, 140.55, and 173.21.
18.
Find x.
Correct Answer
A. 10.64
Explanation
The correct answer is 10.64 because it is the only option that matches the given question of finding x.
19.
Find the measure of angle θ in the figure. Take h = 15 m and d = 30 m.
Correct Answer
C. 25.67°
Explanation
In the given figure, we have a right triangle with one angle of 30°. The side opposite to this angle is h, and the side adjacent to this angle is d. We can use the trigonometric function tangent to find the measure of angle θ. The tangent of angle θ is equal to the ratio of the opposite side (h) to the adjacent side (d). Therefore, we have tan(θ) = h/d. Substituting the given values, we get tan(θ) = 15/30 = 0.5. Taking the inverse tangent of both sides, we find that θ ≈ 25.67°.
20.
A 20-foot ladder is leaning against a wall. The foot of the ladder is 7 feet from the base of the wall. What is the approximate measure of the angle the ladder forms with the ground?
Correct Answer
B. 69.5°
Explanation
The approximate measure of the angle the ladder forms with the ground can be determined using trigonometry. In this case, the ladder forms a right triangle with the wall and the ground. The length of the ladder is the hypotenuse of the triangle, and the distance of the foot of the ladder from the base of the wall is one of the legs. By using the trigonometric function tangent, we can find the angle. The tangent of an angle is equal to the length of the opposite side divided by the length of the adjacent side. In this case, the opposite side is 7 feet and the adjacent side is the length of the ladder. By taking the arctangent of 7 divided by the length of the ladder, we can find the angle. The approximate measure of this angle is 69.5°.