Answer
The lot's dimensions are $70 \times 90$ $ft^{2}$
Work Step by Step
The lot is a rectangle who's only fenced on 3 sides: 1 long-wise and two width-wise. From the information given, we can write the following equations: $$C_{length}(a) = 16a$$ $$C_{width}(b) = 5b$$ $$Perimeter = 2a + 2b = 320$$ $$C_{Total} =C_{length}(a) + 2C_{width}(b) = 2,140$$ where $C_{length}$ represents the costs associated with the fencing of the lot's length, $C_{width}$ represents the costs associated with the fencing of the lot's widths, $C_{Total}$ represents the total costs associated with building the fencing, $a$ represents the length of the lot in feet, and $b$ represents the width of the lot in feet. With this information at hand, we develop the following system of equations: $$16a + 2(5b) = 16a + 10b = 2,140$$ $$2a + 2b = 320$$ To find $a$ and $b$, we can use the substitution method as so: $$2a = 320 - 2b$$ $$a = 160 - b$$ $$THEREFORE$$ $$16(160 - b) + 10b = 2,140$$ $$2,560 - 16b + 10b = 2,140$$ $$2, 560 - 2, 140 = 6b$$ $$\frac{420}{6} = b = 70$$ Having $b$, we now substitute its value in the other equation to find $a$: $$a = 160 - b$$ $$a = 160 - 70 = 90$$ Using these values for $a$ and $b$, we can check and see that the appropriate substitutions are adequate.
