Question

1. list all the various dimensions of the shipping box into which you can pack 30 cubes. there should be five possibilities. watch for dimensions that are the same if you change the position of the shipping box. record them in the following table (the first one is done for you):

dimensions of shipping box	surface area (square inches)
(1 box) 10 inches	(30 boxes) 300 inches	(1 box) 10 inches	$sa = 2lw+2wh + 2lh$ $sa = 2(10\text{ in})(300\text{ in})+2(300\text{ in})(10\text{ in})+2(10\text{ in})(10\text{ in})$ $sa = 12,200\text{ in}^2$

Explanation:

Step1: Factorize 30

The factors of 30 are $1\times30$, $2\times15$, $3\times10$, $5\times6$.

Step2: Case 1 - Dimensions 2, 15, 1

Assume the length $l = 2\times10=20$ inches, width $w=15\times10 = 150$ inches, height $h = 1\times10=10$ inches.
$SA=2lw + 2wh+2lh=2(20)(150)+2(150)(10)+2(20)(10)$
$SA = 6000+3000 + 400=9400$ in²

Step3: Case 2 - Dimensions 3, 10, 1

Let length $l = 3\times10=30$ inches, width $w = 10\times10=100$ inches, height $h=1\times10 = 10$ inches.
$SA=2lw+2wh + 2lh=2(30)(100)+2(100)(10)+2(30)(10)$
$SA=6000 + 2000+600=8600$ in²

Step4: Case 3 - Dimensions 5, 6, 1

Suppose length $l = 5\times10=50$ inches, width $w=6\times10 = 60$ inches, height $h=1\times10=10$ inches.
$SA=2lw+2wh+2lh=2(50)(60)+2(60)(10)+2(50)(10)$
$SA = 6000+1200 + 1000=8200$ in²

Step5: Case 4 - Dimensions 3, 5, 2

If length $l=3\times10 = 30$ inches, width $w=5\times10=50$ inches, height $h=2\times10=20$ inches.
$SA=2lw+2wh+2lh=2(30)(50)+2(50)(20)+2(30)(20)$
$SA=3000+2000 + 1200=6200$ in²

Answer:

Length (in)	Width (in)	Height (in)	Surface area (square inches)
30	100	10	8600
50	60	10	8200
30	50	20	6200