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A triangular parcel of land has sides of lengths 110 feet, 820 feet and 827 feet.

A triangular parcel of land has sides of lengths 110 feet, 820 feet and 827 feet.
1)What is the area of the parcel of land? 
2) If land is valued at 2300 per acre (1 acre is 43,560 feet ), what is the value of the parcel of land? 
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4 Answers

a=110,b=820,c=827 lengths
s=semi-perimeter=1/2(110+820+827)=878.5
s-a=878.5-110=768.5
s-b=878.5-820=58.5
s-c=878.5-827=51.5
Heron's formula for area=√(s(s-a)(s-b)(s-c))=√(878.5(768.5)(58.5)(51.5))=45099.774
1 acre is 1/640 sq mi=52802/640=43560 ft2
Area is 45099.774/43560 acres =1.0353 acres
Value = 2300×1.0353 =2381.30
We have a triangle ABC where AB=c, BC=a, and CA=b
AB=827 ft, BC=110 ft, and CA=820 ft
so c=827, a=110, and b=820
using the law of cosines we have a^2=b^2+c^2-2bc cos(A)
angle A is opposite BC, angle B is opposite AC, and angle C is opposite AB
110^2=820^2+827^2-2*820*827*cos(A)
12,100=672,400+683,929-2*820*827*cos(A)
12,100=1,356,329-1,356,280cos(A)
1,356,280cos(A)=1,356,329-12,100
1,356,280cos(A)=1,344,229
cos(A)=1,344,229/1,356,280
cos(A)=0.9911147
angle A= between 7 degrees 39 minutes and 7 degrees 38 minutes
(0.991099)+(0.991138)/2=0.9911185 (approx. answer)
sine of 7 degrees 39 minutes=0.13312
sine of 7 degrees 38 minutes=0.13283
Theorem: The area of a triangle equals 1/2 the product of the lengths of two sides and the sine of their included angle.
A=(1/2)(820)(827(0.13312)
A=(1/2)(678,140)(0.13312)
A=(339,070)(0.13312)
A=45,136.998 sq ft
A=(339,070)(0.13283)
A=45,038.668 sq ft
45,137/43,560=1.0362 acres
45,038.67/43,560=1.0339 acres
1.0362*$2300=$2383.26 is the value of the land
1.0339*$2300=$2377.97 is the value of the land
if you take the average of the two values you get
$2383.26+$2377.97=$4761.23/2=$2380.62 for the value
Within the limits of "rounding off" this is a right triangle with the two sides being 110 feet and 820 feet in length.
 
Area = (0.5)b*h = (0.5)(110)(820) = 45,100 ft2
 
The land's value is ($2,300 acre-1)*(45,100 ft2)/43560 ft2 acre-1) = $2381
You can get an approximate answer for the area if you treat the triangle as isosceles (820≈827), with base b=110 feet and height h=820 feet. In that case,
A = (1/2) b*h = (1/2) 110*820 = 45100 ft2.
 
You get the exact answer from Heron's formula,
A = sqrt( s*(s-a)*(s-b)*(s-c) )
 
where s is half the perimeter, s = (820+827+110)/2 = 878.5 ft, so that
A = sqrt( 878.5*(878.5-820)(878.5-827)*(878.5-110) ) = 45099 ft² = 1.035 acre,
 
for a value of 2300*1.035=2381.