Financial Mathematics

Question

A rancher is putting up a rectangular cattle pen. His original plan was too expensive, so

he changed to a smaller size, subtracting 𝒙 feet from each of the original boundaries. By

fencing a rectangular space 𝟕𝟔𝟒 𝐲𝐝𝟐 smaller than the original plan, the rancher saves $648 in

fencing materials, which cost $9 per linear foot. How much does he spend?

a. $5,832

b. $6,552

c. $6,876

d. $7,200

Answer 1

Confirming Lexi's answer.

After reading through the problem a couple times, then drawing a diagram, the first thing I would do is convert the 764 yd² to feet squared so that all units are in feet. Since 1 yd² = 9 ft² the new plot is 9(764) ft² smaller. That is 6876 ft².

Use the perimeters to determine expressions for the original cost and the new cost, where L and W represent the length and width of the original plot in feet.

An expression for the original cost is: 9[2L + 2W] = 18L + 18W. This is because each linear foot costs $9.

An expression for the new cost is: 9[2(L-x)+2(W-x)] = 9[2L - 2x + 2W - 2x] = 9[2L + 2W - 4x] = 18L + 18W - 36x. Note that the new dimensions are represented by L - x and W - x, because x feet subtracted from each.

If you subtract the new cost from the original cost you get the amount saved which we know is $648.

18L + 18W - [18L + 18W - 36x] = 648

36x = 648

x = 18

So, 18 feet were subtracted from each dimension of the original plot.

We can use that number to determine an expression for L + W (it just works out that way).

A_originalPlot= LW

A_newPlot = (L - x)(W - x) = LW - Lx - Wx + x²

ΔA = A_originalPLot - A_newPlot = LW - [LW - Lx - Wx + x²] = Lx + Wx - x²

But we know that change in area is 6876.

Lx + Wx - x² = 6876

Or:

(L + W)x - x² = 6876

But we know x = 18.

(L + W)18 - 18² = 6876

L + W = (6876 + 18²)/18 = 400

Now if you go back and look at the expression that represents the new cost (the amount spent):

18L + 18W - 36x

18(L + W) - 36x

18(400) - 36(18)

7200 - 648

$6552

For problems like this, sometimes you just have to start writing stuff down to see where it leads. Do that "stuff" on scratch paper until you see that your work is heading towards a solution. Many times your first efforts will lead down a wrong path and you will have to start over. That is part of the art of problem solving.

Answer 2

Thank you for watching! In my video, I stated that the answer was how much he saved, but I meant spent! Apologies! I hope this helps!

Answer 3

C $6876 spent

saved $648

originally would have spent 6876+648= $6876

= 9 x 764, 764 yd^2 less