Malachi and Talia are refreshing a triangular flower bed at their school. They want to know the area, so they know the amount of material they need. They know two adjoining sides measure 26 feet and 32 feet, and the angle formed between those sides measures 98 degrees. What is the area of their triangular flower bed to the nearest square inch?
We can start by finding the length of the remaining side of the triangle using the Law of Cosines. Let's call this side c.
c² = a² + b² - 2abcos(C), where a and b are the given sides and C is the angle formed between those sides.
c² = 26² + 32² - 2(26)(32)cos(98°)
c ≈ 34.85 feet (rounded to the nearest hundredth)
Now that we have all three sides of the triangle, we can use Heron's formula to find the area of the triangle.
A = √(s(s-a)(s-b)(s-c)), where s is the semi-perimeter of the triangle.
s = (a + b + c)/2
s = (26 + 32 + 34.85)/2
s ≈ 46.92 feet (rounded to the nearest hundredth)
A ≈ √(46.92(46.92-26)(46.92-32)(46.92-34.85))
A ≈ √(46.92*20.92*14.92*12.07)
A ≈ √(19,674.64)
A ≈ 140.21 square feet (rounded to the nearest hundredth)
Since 1 foot is approximately equal to 12 inches, the area in square inches is:
140.21 * (12²) ≈ 201,952.8 square inches
Therefore, the area of their triangular flower bed is approximately 201,952 square inches.