Chapter 9.6, Problem 1MRPS

Solution

i.

To find: How long is the third side of the property?

Hence, length of third side of the property is 680 yards.

Given: Two sides of triangular property are 540 yards and 330 yards, one angle included is $100°$.

Concept Used: Cosine rule: $a^2=b^2+c^2-2bc\cos \angle A$

Calculation:

From the figure:

  

  $\angle A=100°,c=540,b=330$

Using cosine rule:

  $\begin{array}{l}{a}^2=b^2+c^2-2bc\cos \angle A\\ a^2={540}^2+{330}^2-2\times 540\times 330\cos 100°\\ a=\sqrt{291600+108900-(-61888.2105)}\\ a=\sqrt{462388.211}\\ a=680\text{ yards}\end{array}$

Conclusion:

Hence, length of third side of the property is 680 yards.

ii.

To find: How much does the land cost?

Hence, price of land is $45323.5388.

Given: Price of land is $\$2500\text{ per acre (1 acre = 4840 square yards)}$.

Concept Used: Formula of area of triangle is $\frac{1}{2}bc\sin A$.

Calculation:

From the figure:

  

  $\angle A=100°,c=540,b=330$

Area of triangle:

  $\begin{array}{l}A=\frac{1}{2}\times 540\times 330\times \sin 100°\\ A=270\times 330\times \sin 100°\\ A=87746.3708\text{ squares yards}\end{array}$

Now,

  $\begin{array}{l}A=\frac{87746.3708}{4840}\\ A=18.1294155\text{ squares acres}\end{array}$

Price of land is $2500\times 18.1294155=\$45323.5388$

Conclusion:

Hence, price of land is $45323.5388.