The shortest side of a triangular piece of property with area 218,000 ft^2 is on a straight river and requires no fence. How much fence is needed to construct fences on the remaining 2 sides if the angles at the vertices of the property are 41 degrees, 64 degrees, and 75 degrees?
Please help!!! I think it has to do with law of cosines but I'm so lost. Thank you.
Please help!!! I think it has to do with law of cosines but I'm so lost. Thank you.
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Let
A = 75°
B = 64°
C = 41°
It's impossible to determine the area of a triangle given the three angles, since that tells us nothing about the size of the triangle. The law of cosines won't really help either. However, you are given the area.
FYI, the shortest side is always opposite the smallest interior angle, and hence the side opposite to the 41° angle will be the smallest. So, just make sure to plug in correct values when doing these kinds of problems :)
You can find the length of one side using this formula:
A = (a²*sin(B)*sin(C)) / (2*sin(A))
Plug your values in and solve:
218,000 ft² = (a²*sin(64°)*sin(41°)) / (2*sin(75°))
a = 845.11 ft
You can find the length of another side using this formula:
A = ½ab*sin(C)
Plug in your values again:
218,000 ft² = ½(845.11 ft)*b*sin(41°)
b = 786.38 ft
So, your two longest sides are:
a = 845.11 ft
b = 786.38 ft
Which totals up to 1631.49 ft.
You can check this answer if you wish afterwards using the law of sines (to find the remaining side) and Heron's formula, since then you would have found the length of all three sides.
A = 75°
B = 64°
C = 41°
It's impossible to determine the area of a triangle given the three angles, since that tells us nothing about the size of the triangle. The law of cosines won't really help either. However, you are given the area.
FYI, the shortest side is always opposite the smallest interior angle, and hence the side opposite to the 41° angle will be the smallest. So, just make sure to plug in correct values when doing these kinds of problems :)
You can find the length of one side using this formula:
A = (a²*sin(B)*sin(C)) / (2*sin(A))
Plug your values in and solve:
218,000 ft² = (a²*sin(64°)*sin(41°)) / (2*sin(75°))
a = 845.11 ft
You can find the length of another side using this formula:
A = ½ab*sin(C)
Plug in your values again:
218,000 ft² = ½(845.11 ft)*b*sin(41°)
b = 786.38 ft
So, your two longest sides are:
a = 845.11 ft
b = 786.38 ft
Which totals up to 1631.49 ft.
You can check this answer if you wish afterwards using the law of sines (to find the remaining side) and Heron's formula, since then you would have found the length of all three sides.