Question

This shape is made out of rectangles each 4 units tall; the rectangle at the very bottom is 16 units wide; each next rectangle is half the width of the one below it. There are so many rectangles that the finished drawing would not fit on the page, so you can see that the drawing is incomplete. Calculate the area of the entire shape – even the part that is not drawn.

Answer 1

128 units^2

Explanation:

An equation can be written that provides the total area of the rectangles as a function of the rectangle number, which we'll say starts with 1 at the bottom and goes up from there, in whole numbers, to the very top rectangle. The equation would take the form of A(n) = Sum of areas from 1 to n.

I know the equation should be easily derived (but I failed), so I chose to take a faster (for me) path. I set up a spreadsheet that calculates the areas for rectangles from 1 to n, all with height 4 units, but the base is reduced by 1/2 for each increase of n. See the attached spreadsheet image.

As can be seen, the total area increases rapidly from n = 1 to 10. The cumulative area is 127.875 units^2 for 10 rectangles. That's up from 64 units^2 for the 1st (base) rectangle. But as n increases from 10 to 20, the cumulative area only increases from 127.875 to 127.999 units^2. It finally reaches a plateau at 128 units^2. Rectangles beyond this point have a vanishingly small impact on the total area.

Within the limits of my spreadsheet, the total area for all rectangles from 1 to 26 is 128 units^2. Even with 100 rectangles, the area is still 128 units^2.