Application problem on volume of three dimensional figures

The foundation walls in the basement of your new house will be 8 in. thick and 6 ft high. The basement is 24 ft by 32 ft. The floor is 4 in. thick. Under the walls is a 1-ft-wide-by-1-ft-deep footer. Ignoring windows and doors, about how many cubic yards of cement will you need for the walls, floor, and footer?

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Have you find any intermediate results, like the volume of the floor? –  Daan Michiels May 9 '12 at 11:12
Conceded that we have the advantage of the metric system; but otherwise this is a typical problem in the "qualifying exam" for 12-year-olds aspiring to enter the Zurich "Gymnasium". I'm a little surprised. –  Christian Blatter May 9 '12 at 11:35

We've got inches, feet, and yards floating around, but most measurements are in feet, there are no given measurements in yards, and the given measurements in inches are $4\text{ in.}=\frac{1}{3}\text{ ft}$ and $8\text{ in.}=\frac{2}{3}\text{ ft}$, so let's work in feet.

The volume of the floor is $\frac{1}{3}\cdot24\cdot32\text{ ft}^3=\cdots$

Since the problem says "about how many," I'd suppose that we could ignore the corners of the basement (the wall and footer have to extend beyond the dimensions of the floor in order to go around it, making them longer along each side than the edges of the floor), so let's start with that. The perimeter of the floor is $24+32+24+32\text{ ft}=112\text{ ft}$, so the total volume of the walls is $112\cdot\frac{2}{3}\cdot6\text{ ft}^3=\cdots$; the total volume of the footers can be computed similarly.

Adding up these three volumes gives the approximate total volume in cubic feet. Since $3\text{ ft}=1\text{ yd}$, $3^3\text{ ft}^3=27\text{ ft}^3=1\text{ yd}^3$, dividing the number of cubic feet by 27 will give the equivalent number of cubic yards.

If you do need to consider the corners to get a more precise answer, I'd start by considering an overhead diagram of the situation, showing the floor and the wall (or footer—one at a time) around it, and marking off what parts have already been accounted for.

And, as Gerry Myerson's comment implies, marking as accepted the answer that you find best answers your question will make people more likely to answer your questions in the future.

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