Problem 48 - Precalculus Mathematics for Calculus

Question

Two tugboats that are \(120 \mathrm{ft}\) apart pull a barge, as shown. If the length of one cable is \(212 \mathrm{ft}\) and the length of the other is \(230 \mathrm{ft}\), find the angle formed by the two cables.

Step-by-Step Solution

Answer

The angle formed by the two cables is approximately 31.58°.

Step 1: Understand the Problem Setup

We have two tugboats and a barge forming a triangle. The sides of the triangle are the lengths of the cables (212 ft and 230 ft) and the distance between the tugboats (120 ft). We need to find the angle between the two cables.

Step 2: Identify the Formula

We use the Law of Cosines, which relates the lengths of the sides of a triangle to the cosine of one of its angles. For sides \(a\), \(b\), \(c\), and angle \(C\) opposite \(c\), it's given by: \[ c^2 = a^2 + b^2 - 2ab \cos(C) \] In this case, \(c = 120\) ft, \(a = 212\) ft, and \(b = 230\) ft.

Step 3: Apply the Law of Cosines

Substitute the values into the Law of Cosines formula: \[ 120^2 = 212^2 + 230^2 - 2(212)(230) \cos(C) \] This becomes \[ 14400 = 44944 + 52900 - 97520 \cos(C) \]

Step 4: Solve for \(\cos(C)\)

Rearrange the equation to solve for \(\cos(C)\): \[ 97520 \cos(C) = 44944 + 52900 - 14400 \] \[ 97520 \cos(C) = 83444 \] \[ \cos(C) = \frac{83444}{97520} \]

Step 5: Calculate the Cosine

Calculate \(\cos(C)\) to find: \[ \cos(C) \approx 0.8559 \]

Step 6: Determine the Angle

Use the inverse cosine function to find \(C\):\[ C = \cos^{-1}(0.8559) \] Calculate to find:\[ C \approx 31.58^\circ \]

Key Concepts

Triangle Geometry

When dealing with triangle geometry, it's essential to understand the properties that define a triangle. A triangle consists of three sides and three angles. In this exercise, the sides are the lengths of the cables and the distance between the tugboats. The specific triangle type we encounter is a scalene triangle, where all three sides have different lengths. This results in all the angles being different as well.
Understanding triangle geometry allows us to apply specific rules and formulas, like the Law of Cosines, to solve for unknown angles or sides. Remember, to determine any angle or side in a triangle, at least one side and two additional measurements (either sides or angles) must be known. Knowing this, we can confidently leverage triangle geometry to find the angle between two sides, as in this exercise.

Inverse Trigonometric Functions

Inverse trigonometric functions are critical when solving for angles in triangles using known side lengths. They help us find angles when we already know the values of trigonometric ratios like sine, cosine, or tangent. In this problem, we used the inverse cosine function because we needed to find the angle formed by the two cables given the cosine value.
Essential information to keep in mind about inverse trigonometric functions is:

• The result of an inverse trigonometric function is an angle.
• The function reflects the relationship between the side lengths and angles within the triangle.
• The domain and range of these functions are limited depending on the angle and ratio.

In the specific case from the exercise: once we found the value of \(\cos(C)\), using the inverse cosine function \(\cos^{-1}(x)\) allowed us to calculate the actual angle \(C\). Calculators or software often have these functions built-in to easily perform these calculations, resulting in a practical solution for finding angle measurements precisely.

Solving for Angles

Solving for angles in triangles involves identifying the appropriate formula and applying it correctly. In our exercise, after setting up our triangle with known sides, we used the Law of Cosines to derive the cosine of the angle we wanted to find.
Key steps in solving for an angle using this method include:

• Substituting known side lengths into the Law of Cosines formula, \[c^2 = a^2 + b^2 - 2ab \cos(C)\].
• Rearranging the equation to solve for \cos(C)\.
• Using the inverse cosine function to determine the angle from the cosine value.

Working through this sequence ensures we accurately solve for unknown angles, leveraging both geometry and trigonometry and providing deeper insights into the spatial relationships between objects, as represented by our triangle. In practical scenarios like tugboats and cables, determining these angles is crucial for safe and effective navigation and positioning.