Math Problem Statement

A triangular swimming pool measures

4545

ft on one side and

32.232.2

ft on another side. The two sides form an angle that measures

40.740.7degrees°.

How long is the third​ side?

Question content area bottom

Part 1

The length of the third side is

enter your response here

ft.

​(Simplify your answer. Type an integer or a decimal. Round to the nearest tenth if​ needed.)

Solution

To calculate the length of the third side of the triangle, we can use the Law of Cosines. The formula is:

c2=a2+b22abcos(C)c^2 = a^2 + b^2 - 2ab \cdot \cos(C)

Where:

  • aa and bb are the known sides of the triangle
  • CC is the angle between the two known sides
  • cc is the third side we want to calculate

Given:

  • a=45fta = 45 \, \text{ft}
  • b=32.2ftb = 32.2 \, \text{ft}
  • C=40.7C = 40.7^\circ

Step 1: Apply the formula

c2=452+32.2224532.2cos(40.7)c^2 = 45^2 + 32.2^2 - 2 \cdot 45 \cdot 32.2 \cdot \cos(40.7^\circ)

Step 2: Compute values

  1. 452=202545^2 = 2025
  2. 32.22=1036.8432.2^2 = 1036.84
  3. Compute 24532.2=28982 \cdot 45 \cdot 32.2 = 2898
  4. cos(40.7)0.7602\cos(40.7^\circ) \approx 0.7602

Substituting: c2=2025+1036.8428980.7602c^2 = 2025 + 1036.84 - 2898 \cdot 0.7602

c2=2025+1036.842201.3776c^2 = 2025 + 1036.84 - 2201.3776

c2860.4624c^2 \approx 860.4624

Step 3: Take the square root

c=860.462429.3ftc = \sqrt{860.4624} \approx 29.3 \, \text{ft}

Final Answer:

The length of the third side is approximately 29.3 ft.


Would you like more details or assistance with related questions?
Here are five related questions for exploration:

  1. What is the formula for the Law of Cosines, and when is it used?
  2. How do you calculate the area of a triangle with two sides and the included angle?
  3. What are some real-world applications of the Law of Cosines?
  4. How does the Law of Cosines generalize the Pythagorean theorem?
  5. How can you calculate the remaining angles in the triangle after finding the third side?

Tip: Always keep your calculator in degree mode when working with angles measured in degrees.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Triangle Geometry
Law of Cosines

Formulas

c² = a² + b² - 2ab * cos(C)

Theorems

Law of Cosines

Suitable Grade Level

Grades 10-12