A carpenter is adding a sun room to the back of a house. If the width of the room is 15 feet, the distance from the house to either obtuse corner of the room is 13 feet, and the distance from the house to the point of the room is 18 feet, then angle = _____degrees. Round to one decimal place.
To find the angle, we can use the law of cosines. The law of cosines states that, in a triangle with sides of lengths a, b, and c, and opposite angles A, B, and C, the following equation holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, we can label the sides of the triangle using the given information:
a = 13 feet (distance from the house to either obtuse corner)
b = 18 feet (distance from the house to the point of the room)
c = 15 feet (width of the room)
We want to find the angle C, which is the angle opposite to the side of length c.
Rearranging the equation, we have:
cos(C) = (a^2 + b^2 - c^2) / (2ab)
Now, we can substitute the values into the equation:
cos(C) = (13^2 + 18^2 - 15^2) / (2 * 13 * 18)
Simplifying the equation:
cos(C) = (169 + 324 - 225) / (468)
cos(C) = 268 / 468
cos(C) ≈ 0.5726
To find the angle C, we can take the inverse cosine (or arccos) of 0.5726 using a calculator:
C ≈ arccos(0.5726)
C ≈ 55.7 degrees
Therefore, the angle is approximately 55.7 degrees when rounded to one decimal place.