Problem 43 - Algebra 2

Question

Sailing Buoys are located in the sea at points \(A, B,\) and \(C . \angle A C B\) is a right angle. \(A C=3.0 \mathrm{mi}, B C=4.0 \mathrm{mi},\) and \(A B=5.0 \mathrm{mi}\) A ship is located at point \(D\) on \(\overline{A B}\) so that \(m \angle A C D=30^{\circ} .\) How far is the ship from the buoy at point \(C ?\) Round your answer to the nearest tenth of a mile.

Step-by-Step Solution

Answer

The ship's distance from the buoy at point C is 1.5 mi.

Step 1: Identify the triangles

Identify two triangles from the problem, triangle ACB, which is a right triangle, and triangle ACD, which is a 30-60-90 triangle.

Step 2: Find the length of AD

Because triangle ACD is a 30-60-90 triangle, the sides are in the ratio of \(1: \sqrt {3}: 2\). The side opposite the 30° angle is half the length of the hypotenuse (AD = AB/2). Substituting given values, AD = 5.0 mi / 2 = 2.5 mi.

Step 3: Calculate the length of CD

Next, calculate the length of CD which is the distance from the ship to buoy C. In the right triangle ACB, knowing AC = 3.0 mi and AB = 5.0 mi, CB can be calculated by using the Pythagorean theorem, \(AB^2 = AC^2 + CB^2\). Rearranging and substituting given values, we find \(CB = \sqrt{AB^2 - AC^2} = \sqrt{(5.0 \,mi)^2 - (3.0 \,mi)^2} = 4.0 \,mi\). Then, since CD = CB - BD, substituting calculated and given values we find \(CD = 4.0 \,mi - 2.5 \,mi = 1.5 \,mi\).

Step 4: Round the answer:

As per the problem's instructions, round your answer to the nearest tenth of a mile. In this case no rounding is required as our answer is already to the nearest tenth of a mile.

Key Concepts

Right Triangle

A right triangle is a triangle in which one of the angles is exactly 90 degrees. This type of triangle has some unique properties that make it special. The side opposite the right angle is called the hypotenuse, and it is the longest side of the triangle. The other two sides are traditionally known as the legs.
Understanding right triangles is fundamental in trigonometry as they serve as the basis for defining trigonometric ratios such as sine, cosine, and tangent. In the context of the problem, triangle ACB forms a right triangle with angle ACB being the right angle. Knowing the properties of a right triangle helps in solving various geometric problems.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, particularly applicable to right triangles. It states that in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the hypotenuse. Mathematically, it is expressed as:

• \(a^2 + b^2 = c^2\)

where \(c\) represents the hypotenuse and \(a\) and \(b\) represent the other two sides.
This theorem is instrumental in finding the length of one side of a right triangle when the lengths of the other two are known. In our exercise, we used the Pythagorean Theorem to solve for side BC in triangle ACB, confirming its length as 4 miles, given the other sides were 3 and 5 miles. This kind of calculation is particularly useful in various fields like engineering and physics, where the relationships between dimensions need to be computed accurately.

30-60-90 Triangle

A 30-60-90 triangle is a special type of right triangle that has angles of 30, 60, and 90 degrees. This attribution leads to a specific ratio among the lengths of its sides:

• The shortest side is opposite the 30-degree angle.
• The side opposite the 60-degree angle is \(\sqrt{3}\) times the length of the shortest side.
• The hypotenuse is twice the length of the shortest side.

In our exercise, triangle ACD is identified as a 30-60-90 triangle given that angle ACD is 30 degrees. This property allows us to easily determine that the piece of line AD is half of the entire line AB, simply calculated as 2.5 miles.
Recognizing a 30-60-90 triangle quickly helps to simplify complex problems without the need for additional trigonometric functions, making it a handy tool for efficient problem-solving in both math and real-world applications.