Border Around a Garden

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mathxyz

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Jul 8, 2005
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A landscaper, who just completed a rectangular flower garden measuring 6 feet by 10 feet, orders 1 cubic yard of premixed cement, all of which is to be used to create a border of uniform width around the garden. If the border is to have a depth of 3 inches, how wide will the border be?
(1 cubic yard = 27 cubic feet)
 
The volume of the cement is 27 cubic feet, and the depth is 1/4 feet, so the area on top is (27 ft^3) / (1/4 ft) = 27 * 4 ft^2 = 108 ft^2.

Let w = width of border

Then the area of the border is ((2w + 6) * (2w + 10) - 6 * 10) ft^2.
 
okay

I can now solve the question. I just don't know how to set up the proper equation that leads me to the answer.
 
Alex, confess. Did YOU know the proper formula when you started this problem, or did you make a diagram, label some parts, and derive the formula for yourself?
 
Hehe, no, I don't have formulas for border problems lying around or memorized. I made a diagram, labeled some parts, and derived the formula myself.

Do you see where those formulas come from, mathxyz?
 
No...

My reply to alexflury:

Your question:

"Do you see where those formulas come from, mathxyz?"

No, that's my problem. I have not been able to, for some reason or another, form proper equations to use as a guide for answering word problems. Trust me, if I could learn this skill, I would love math even more.
 
The old border around a garden problem is a little tricky. Hardly anyone gets it on first exposure. It has two elements that make it a little more difficult.

1) It is very common to forget that the border is on BOTH sides of the length and the width

2) You can find the area most easily by NOT doing it directly. Calculating the entire area and then subtracting the part you don't want, is much simpler.

Garden Measure 6 ft x 10 ft
Garden + Border Measure: (Border + 6 ft + Border) x (Border + 10 ft + Border) = (6 ft + 2*Border) x (10 ft + 2*Border)

See how the border needs to happen TWICE?

Area of the border: [Area of the whole thing] LESS [the area of the Garden]

[(6 ft + 2*Border) x (10 ft + 2*Border)] - [(6 ft) x (10 ft)]
 
Memorize this:

l = garden's length, w = garden's width, b = border's width

Area of border = b(b + l + w)
 
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