In honor of Father’s Day, this weekend challenge is inspired by my dad’s hobby business. He sells wooden flags, like the one above, on Etsy. Basically, he finds discarded pallets (this is a pallet), chops ’em up into pieces of the size he needs, and mounts them.

As usual, these challenge questions are not really SAT questions, they just require you to work the same muscles that you need to exercise to do well on the SAT. Consider them cross-training. Here we go.

The creation of one of these flags requires 13 solid strips of wood that are 1 inch thick, 2 inches wide, and 36 inches long. If my dad makes as many flags as possible out of a pallet with 10 usable slats that are one inch thick, 7 inches wide, and 40 inches long, what is the volume of the leftover scrap in cubic inches? Disregard sawdust loss and any part of the pallet that was not originally deemed usable.

Put your answers in the comments. First correct (non-anonymous) answer gets a Math Guide. Good luck, and don’t forget to wish the fathers in your life a happy Father’s Day!

UPDATE: Congratulations to Shahriar, who got it first. I’ll be in touch shortly to get your shipping information, Shahriar, so I can send you your Math Guide.

Solution below the cut.
I realized after I posted this that my measurements weren’t very realistic at all, but whatever. It’s the math that’s important, not whether my fictional pallet measures up to industry standards.

The first step to solving this problem is to recognize that by making the thickness of the slats 1 inch, I went easy on you: the value of the volume of wood left over will be the same as the value of the area of its front-facing face. In other words, you can basically treat this as a two-dimensional problem even though I’m asking for volume. You’re welcome.

How many usable strips of wood can be made from one pallet slat? 3 of them. And once those 3 usable strips are cut away, each slat produces 36+28=64 in3 of scrap.

Of course, there are 10 such slats, so you’re looking at something like this:
Oh but wait. That produces 30 usable slats. Which is only enough to make 2 flags, since each flag requires 13 strips. I guess 4 of those strips become unusable scrap, too. 🙁

So really, the situation is that each slat produces at least 64 in3 of scrap. We have to add the volume of the 4 unnecessary strips (32×2=72 in3 each) to our final volume, since they’re not actually used in the making of flags.

(10 slats × 64 in3 scrap) + (4 strips × 72 in3 scrap) = 928 in3 of scrap

abhijitjain1995 says:

928 is it 🙂 (but i do have a book ) m/ is it rite ? and how can i share my file where i solved the question ?

abhijitjain1995 says:

i am typing the answer here – one flag – 13*1*2*36 = 936
total vol = 10 * 1*7*40 = 2800 .
possible # of flags = 2 (928/936) . And so we are left wid 928 as our answer . tell me if this is wrong mike .

RexT says:

nevermind, didnt see as many flags as possible, it is 928