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Geometry
Word Problems: Sections: Introduction, Basic examples, Triangle formulas, Complex examples, The Box Problem & the Goat Problem, Max / Min problems The following are a couple of "classic" types of exercises. Almost all students eventuall see one or both of these, in some form.
You have a large piece of cardboard, but you don't have enough cardboard to make a mistake and try again, so you'll have to get it right the first time. You will be forming the box by cutting out a large square, and then cutting out the twoinch squares from the corners that will allow you to fold up the edges to make a twoinchdeep box. What should be the dimensions of the large square? (Ignore the top of the box: you'll just make another open box, slightly larger, turn it upside down, and slip it over the first box to make the "top".) Visually, this is what I'm doing:
I will use "x" to stand for both the length and the width of the original square of cardboard in the computations that follow. The value of x is what I'm looking for. Looking at the picture, I can see that the width of the bottom of my box will be x – 2 – 2; that is, the final width of the box will be the sheet's original x inches, minus two inches on either side because of the portions that I'll be losing to the flaps I'll be folding up. Then the width of the bottom of my box is going to be x – 4. By the same reasoning, the length will also be x – 4. Since the depth of my box is 2 (that's the whole point of the twoinchbytwoinch squares that I'm cutting out of the corners), I can write the formula for the volume of this box as: volume
= (length)(width)(depth)
= 512
cubic inches
I can ignore the extraneous negative result. Then x = 20 inches, and the answer is: The large piece of cardboard should be twenty inches square. Warning: "Twenty inches square" is not the same as "twenty square inches". "Twenty inches square" means "twenty inches on a side", for a total area of four hundred square inches. "Twenty square inches" is just that: twenty square inches. Copyright © Elizabeth Stapel 20002011 All Rights Reserved
What formula could they be expecting me to use for this? How on earth would I answer this? When in doubt, it can be helpful to draw a picture, so let's see what I can find: The goat is tied at the upper righthand corner with the dot. Now what? Well, on the nearest two sides, the goat can go this far: This is threequarters of a circle with radius 8 (the full length of the rope). The goat can walk around the far corners as far as the rope will allow. Along the top side, five meters of rope will be stretched along the side, leaving another three meters "in play"; along the righthand side, four meters will have been used, leaving another four meters: These new areas are both quartercircles, one with radius 3 and the other with radius 4. And there will be no way for the goat to reach the opposite corner or "overlap" in grazing, because he's out of rope. So these three partial circles mark off the total grazing area. I know the formula for the area of a circle. To find, say, the area of 3/4 of a circle, I'd just multiply the total area by 3/4. So the total area is: (3/4)(pi)(8^{2})
+ (1/4)(pi)(3^{2}) + (1/4)(pi)(4^{2})
I'm supposed to give my answer in terms of the nearest whole unit, so I'll plug the above into my calculator, and round: The goat can graze about 170 square meters of grass. Note how drawing the picture allowed me to "see" what I needed to do, immediately simplying my work. If you have "no idea" what to do, try drawing a picture. (There is an extension of this exercise, which has the goat's areas overlapping at that opposite corner. The exact solution involves trigonometry or calculus.) << Previous Top  1  2  3  4  5  6  Return to Index Next >>



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