A farmer wants to build a small, rectangular pen next to a barn. She has 55 feet of fence available and needs the length of the pen to be three times the
width. If the barn forms the fourth side of the pen as shown in the figure
below, what dimensions (length and width) should the pen be so that all 55 feet
of fence are used with the other three sides? *You must use a variable to
write an equation, show your work, and label your correct answers with the
correct units for full credit.
Equation: _

Width _

Length _

Units for both width and length _

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!

5x+2x=49

7x=49

x=7. 35 apples