The following exercise uses Heron's formula One triangular plot has sides 5 yards, 6 yards, and 7 yards. Another has sides 3 yards, 6 yards, and 7 yards. Find the area of each plot. (Round your answers to one decimal place.) first plot second plat Which plot endloses the larger area? first plot second plat

Accepted Solution

Answer:[tex]\text{Area of 1st plot}\approx 14.7\text{ yards}^2[/tex][tex]\text{Area of 2nd plot}\approx 8.9\text{ yards}^2[/tex]1st plot encloses the larger area.Step-by-step explanation:We have been given the sides of two triangles. We are asked to find the area of given triangles using Heron's formula.The area of a triangle with sides a, b and c would be:[tex]\text{Area of }\Delta =\sqrt{S(S-a)(S-b)(S-c)}[/tex], where S is the semi-perimeter of triangle.[tex]S=\frac{5+6+7}{2}=\frac{18}{2}=9[/tex]Substitute given side lengths:[tex]\text{Area of 1st plot}=\sqrt{9(9-5)(9-6)(9-7)}[/tex][tex]\text{Area of 1st plot}=\sqrt{9(4)(3)(2)}[/tex][tex]\text{Area of 1st plot}=\sqrt{216}[/tex][tex]\text{Area of 1st plot}=14.6969\approx 14.7[/tex]Therefore, the area of 1st plot would be 14.7 square yards.[tex]S=\frac{3+6+7}{2}=\frac{16}{2}=8[/tex]Substitute given side lengths:[tex]\text{Area of 2nd plot}=\sqrt{8(8-3)(8-6)(8-7)}[/tex] [tex]\text{Area of 2nd plot}=\sqrt{8(5)(2)(1)}[/tex][tex]\text{Area of 2nd plot}=\sqrt{80}[/tex][tex]\text{Area of 2nd plot}=8.9442\approx 8.9[/tex]Therefore, the area of 2nd plot would be 8.9 square yards.Since area of first plot is greater than 2nd plot, therefore, 1st plot encloses the larger area.