^{2}/640=43560 ft

^{2}

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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Michael F. | Mathematics TutorMathematics Tutor

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=5280^{2}/640=43560 ft^{2}

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 ft^{2})/43560 ft^{2} 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 ft^{2}.

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.

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