How many DVDs can fit in the DVD storage box based on the measurements given?

Question

If each DVD measures 5 3/18'' x 7 1/2'' x 1/2'', and the storage box measures 15'' tall and has a base measuring 2' x 3' x 7'', how many DVDs can fit in the storage box?

Answer 1

Since the DVDs are rigid, this is not a simple math problem. The temptation would be to divide the volume of the box (24" x 36" x 7") by the volume of each DVD (5 3/18" x 7 1/2" x 1/2") and come up with a number (312-ish), but you would never be able to fit that many in the box.

 

Depending on how you align the sides of the box (W=24", H=36", D=7") with the sides of the DVDs (A=5 3/18", B=7 1/2", C=1/2") you will get different results.

 

Consider DVD dimension B (7 1/2"), it cannot fit into box dimension D (7"). It can only map to W or H.

• FLOOR(W/B) = 3
• FLOOR(H/B) = 4

 

The FLOOR() function takes the next lower integer number, throwing away the remainder. This says we can fit 3 DVDs along box side W (which is 24") when we align it with DVD side B (which is 7 1/2"), and 4 if we use box side H (which is 36").

 

Similarly for dimension A we have

• FLOOR(W/A) = 4
• FLOOR(H/A) = 6
• FLOOR(D/A) = 1

 

And dimension C

• FLOOR(W/C) = 48
• FLOOR(H/C) = 72
• FLOOR(D/C) = 14

 

Now we have to choose the mapping that yields the highest number of DVDs, noting that we must pick one of each [W,H,D] and [A,B,C]. 

 

The possibilities:

• W/A x H/B x D/C = 4x4x14 = 224
• H/A x W/B x D/C = 6x3x14 = 252
• D/A x W/B x H/C = 1x3x72 = 216
• D/A x H/B x W/C = 1x4x48 = 192

 

The mapping H/A x W/B x D/C yields the highest number of DVDs, 252.