NAPT - Navy Advanced Programs Geometric and Trigonometric Principles Questions and Answers — Questions and Answers

Question 1: A ladder is placed against a vertical wall such that its base is 7 feet away from the wall and its top reaches a window 24 feet high. What is the length of the ladder?

31 feet
25 feet (Correct answer)
17 feet
23 feet

Correct answer: 25 feet

The ladder, the wall, and the ground form a right-angled triangle. The length of the ladder is the hypotenuse. According to the Pythagorean theorem, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b): a² + b² = c². In this case, a = 7 feet and b = 24 feet. So, 7² + 24² = c², which is 49 + 576 = c². This simplifies to 625 = c². Taking the square root of both sides gives c = 25. Therefore, the length of the ladder is 25 feet.

Question 2: In a 30-60-90 right triangle, which of the following statements is true regarding the lengths of its sides?

The hypotenuse is twice the length of the longer leg.
The side opposite the 60° angle is twice the length of the side opposite the 30° angle.
The two legs are equal in length.
The hypotenuse is twice the length of the shorter leg. (Correct answer)

Correct answer: The hypotenuse is twice the length of the shorter leg.

In a 30-60-90 triangle, the sides are in a specific ratio. The side opposite the 30° angle is the shortest leg (let's call its length 'x'). The hypotenuse is always twice the length of the shorter leg (2x). The side opposite the 60° angle (the longer leg) is √3 times the length of the shorter leg (x√3). Therefore, the correct statement is that the hypotenuse is twice the length of the shorter leg.

Question 3: A communications tower casts a shadow that is 75 meters long when the sun's angle of elevation is 45°. What is the height of the tower?

75 meters (Correct answer)
50 meters
100 meters
75√2 meters

Correct answer: 75 meters

This scenario forms a right triangle with the tower as the vertical side, the shadow as the horizontal side, and the angle of elevation being 45°. The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side (tan(θ) = opposite/adjacent). Here, θ = 45°, the adjacent side (shadow) is 75m, and the opposite side is the tower's height (h). So, tan(45°) = h / 75. Since tan(45°) = 1, the equation becomes 1 = h / 75. Therefore, h = 75 meters. Alternatively, a right triangle with a 45° angle is an isosceles right triangle, meaning the two legs are equal in length.

Question 4: A circular radar screen has a circumference of 12π inches. What is the area of the screen?

144π square inches
24π square inches
36π square inches (Correct answer)
12π square inches

Correct answer: 36π square inches

First, find the radius from the circumference. The formula for circumference is C = 2πr. Given C = 12π, we can solve for r: 12π = 2πr, which simplifies to r = 6 inches. Next, use the radius to find the area. The formula for the area of a circle is A = πr². Substituting the radius we found: A = π(6)² = 36π square inches.

Question 5: What is the exact value of sin(60°)?

1/2
√2/2
√3/3
√3/2 (Correct answer)

Correct answer: √3/2

The value of sin(60°) is a standard trigonometric identity derived from a 30-60-90 triangle. In such a triangle, if the side opposite the 30° angle is 1, the hypotenuse is 2, and the side opposite the 60° angle is √3. Sine is defined as the ratio of the opposite side to the hypotenuse (SOH). For the 60° angle, the opposite side is √3 and the hypotenuse is 2. Therefore, sin(60°) = √3/2.

Question 6: Which of the following regular polygons has a sum of interior angles equal to 1080°?

Hexagon (6 sides)
Decagon (10 sides)
Nonagon (9 sides)
Octagon (8 sides) (Correct answer)

Correct answer: Octagon (8 sides)

The formula to find the sum of the interior angles of a polygon with 'n' sides is (n - 2) * 180°. We need to find which polygon's 'n' results in 1080°. We can set up the equation: (n - 2) * 180 = 1080. Divide both sides by 180: n - 2 = 1080 / 180, which simplifies to n - 2 = 6. Add 2 to both sides: n = 8. A polygon with 8 sides is an octagon.

NAPT - Navy Advanced Programs Geometric and Trigonometric Principles Questions and Answers

A ladder is placed against a vertical wall such that its base is 7 feet away from the wall and its top reaches a window 24 feet high.
What is the length of the ladder?