NAPT - Navy Advanced Programs Geometric and Trigonometric Principles 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?

Accepted Answer

25 feet

Answer

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?

Accepted Answer

The hypotenuse is twice the length of the shorter leg.

Answer

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?

Accepted Answer

75 meters

Answer

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?

Accepted Answer

36π square inches

Answer

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°)?

Accepted Answer

√3/2

Answer

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.

NAPT - Navy Advanced Programs Test Practice Test

NAPT - Navy Advanced Programs Test Practice Test

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