a house is built on a triangle plot of land. two sides of the plot are 160 feet long and they meet at an angle of 85 degrees. if a fence is to be built around the property, how much fencing material is needed
To find out how much fencing material is needed for the triangle plot of land, we need to calculate the length of the third side.
Given that two sides of the plot are 160 feet long and they meet at an angle of 85 degrees, we can use the law of cosines to find the length of the third side.
The law of cosines states:
c^2 = a^2 + b^2 - 2ab * cos(C)
Where:
c is the length of the third side
a and b are the lengths of the other two sides
C is the angle between the two sides we know
In this case, the angle C is the angle between the two sides of 160 feet, which is 85 degrees.
Let's substitute the values into the formula and solve for c:
c^2 = 160^2 + 160^2 - 2(160)(160) * cos(85°)
c^2 = 25600 + 25600 - 2(160)(160) * cos(85°)
c^2 = 51200 - 51200 * cos(85°)
c^2 ≈ 51200 - 51200 * 0.087
c^2 ≈ 51200 - 4454.4
c^2 ≈ 46745.6
Taking the square root of both sides gives us:
c ≈ sqrt(46745.6)
c ≈ 216.01
Therefore, the length of the third side is approximately 216.01 feet.
Now, to calculate the amount of fencing material needed, we add up the lengths of all three sides:
160 + 160 + 216.01 ≈ 536.01 feet
So, approximately 536.01 feet of fencing material is needed to enclose the triangle plot of land.
To calculate the amount of fencing material needed for the triangular plot of land, we need to find the length of the third side.
Since we have two sides of the triangle and the angle between them, we can use the Law of Cosines to find the length of the third side:
c^2 = a^2 + b^2 - 2ab cos(C)
Where:
c is the third side (length we want to find)
a and b are the two known sides (160 feet)
C is the angle between the two known sides (85 degrees)
Plugging in the values:
c^2 = 160^2 + 160^2 - 2(160)(160) cos(85)
c^2 = 25600 + 25600 - 51200 cos(85)
Now we can calculate the length of the third side (c):
c = √(25600 + 25600 - 51200 cos(85))
c ≈ √(51200 - 51200 cos(85))
c ≈ √(51200 - 32118.66)
c ≈ √19081.34
c ≈ 138.16 feet
Now, to find the amount of fencing material needed, we add the lengths of the three sides of the triangle:
160 + 160 + 138.16 = 458.16 feet
Therefore, approximately 458.16 feet of fencing material is needed to enclose the triangular plot of land.