Problem Solving?

Question

A farmer has three fields. One field is an equilateral triangle, one field is a circle, and one field is a square. The square field is 75% larger in area than the triangular field, and 50% larger in area than the circular field. In order to completely fence all three of the fields, exactly 4000 metres of fencing is required.

What is the total area of all three fields, in square metres?

Answer 1

Let s denote the length of a side of the equilateral triangle, a the length of a side of the square, and r the radius of the circle.

We are trying to find

A=\sqrt{3}/4 s^2+a^2+\pi r^2 (*)

subject to

a^2=7/4 \sqrt{3}/4 s^2 (1)
a^2=3/2 \pi r^2 (2)
3s+4a+\2 pi r=4000 (3)

(1) implies that s=4a/(3^(1/4) \sqrt{7})
(2) implies that r=a \sqrt{2}/\sqrt{3 \pi}

Subsituting these values into (3) and simplifying gives us

2/21 a (42+6*3^(3/4) \sqrt{7}+7 \sqrt{6 \pi})=4000

so

a=42000/(42+6*3^(3/4) \sqrt{7}+7 \sqrt{6 \pi})

Now subsituting all of these values into (*) gives us

A=\sqrt{3}/4 s^2+a^2+\pi r^2
=47a^2/21
=3948000000/(42+6*3^(3/4) \sqrt{7}+7 \sqrt{6 \pi})^2
\approx 334888 m^2

Be sure I used the area formulas correctly! :)

Answer 2

We assume that the 3 fields are not adjacent, so the fencings are around the perimeters of the 3 fields.

Equilateral triangle, of x meters per side:
Area, A1 = (1/2)(x)(x)sin(60deg) = 0.433x^2 sq.m.
Perimeter, P1 = 3x m.

Circle, of radius y meters:
A2 = pi(y^2)
P2 = 2pi(y)

Square, of z meters per side:
A3 = z^2
P3 = 4z

A3 = 1.75(A1) ------(1)
So,
z^2 = 1.75(0.433 x^2)
z = (0.870488)x
Or, x = z/(0.870488) = (1.148781)z ----**

A3 = 1.50(A2) ------(2)
So,
z^2 = 1.50(pi y^2)
y = (0.460659)z ----**

P1 +P2 +P3 = 4000 ----(3)
3x +2pi(y) +4z = 4000
Substitutions,
3(1.148781)z +2pi(0.460659)z +4z = 4000
(10.340749)z = 4000
z = 386.82 meters

And so,
x = (1.148781)(386.82) = 444.37 m.
y = (0.460659)(386.82) = 178.19 m.

Therefore, total area of all fields is
A = A1 +A2 +A3
A = (0.433)*(444.37)^2 +pi*(178.19)^2 +(386.82)^2
A = 334,883 sq. m. -------answer.

Answer 3

Hi, I make it 334887.03 sq m but this is by a horrendously complicated method, will keep working to see if I can find an easier way

Answer 4

tringle area = 1/2 x^2
circle = pie * r^2
square = y^2

y^2= 1.75 (0.5 x^2)
y = sqrt(1.75/2)*x
x = y/sqrt(0.875)

y^2 = 1.5*pie* r^2
r = y/sqrt(1.5*pie)

cricumference:
2*pie*r + 4*y + 3x = 4000
2*pie(y/sqrt(1.5*pie)) + 4y + 3y/sqrt(0.875) = 4000
solve for y = ...

substitue to find x= y/sqrt(0.875)
substitue to find r = y/sqrt(1.5*pie)

Good luck

Answer 5

346169.1145 sq. mt

Answer 6

334887.03