I'm rehabbing a cabin where the total outside surface area is 578 sq.ft. I want to have 3" strips of plywood every 1 foot and put rigid foam in between the strips so I can have something to screw the sheet metal to. So to cut a 4x8 sheet of plywood lengthwise into 3" strips I would want 3.2" strips so I would cut it into 15 equal strips. The rigid foam would be cut lengthwise into 5 pieces to get 9.6" wide strips. That way I won't have any waist. So how do I calculate how many 4x8 sheets of the plywood and and how many 4x8 sheets of foam? Each 3.2" plywood and each 9.6" foam would be 12.8" wide. So I'm thinking:
x = no. of sheets of plywood
y = no. of sheets of foam
x/(3.2 * 96) + y/(9.6 * 96) = 578*144
(96" is the length of each strip)
Is this a related rate problem? It's been 20+ years since I took Calculus.
Edit: I think it would just be:
3.2/12.8 = 0.25
9.6/12.8 = 0.75
plywood covers 25% of surface
foam covers 75% of surface
578 * 0.25 = 144sq/32 = 4 1/2 sheets
578 * 0.75 = 433/32 = 13 1/2 sheets
A strip of plywood and a strip of foam are 12.8 inches wide. Let $W$ be the width of the cabin in feet (width of front + side + side + back) and $L$ the height in feet. To go one time around the cabin (width wise) we need
$$\dfrac{12W}{12.8}$$ strips of plywood and foam. Now that we've gone around once we need to go up again. Each strip is 8 feet long so we need
$$\dfrac{12L}{12(8)}$$
strips to go up the cabin (length/height wise) one time. Multiplying these we need
$$P = \dfrac{12 \times W \times L}{12.8(8)}$$
pairs of strips. So we will need $\dfrac{P}{15}$ sheets of plywood and $\dfrac{P}{5}$ sheets of foam.
Now $W \times L$ is the surface area so using the 578 square feet you've given we need. 67.734375 pairs of strips from each. So that's $4.52$ sheets of plywood and $13.54$ sheets of foam, as you have. I'd round these up and add an extra or two for mistakes.
One other thing to consider is for certain widths you won't be able to finish with a complete pair. For example if the width were $28$ inches you would do plywood, foam, plywood would be $3.2 + 9.6 + 3.2 = 16$ inches. You can't add another 9.6 width piece of foam and 3.2 inch plywood as you would have $16 + 9.6 + 3.2 = 28.8$ inches. In this case you might just want to cut the foam to $12$ inches so it'll add up to 28, but then you'll need more foam and slightly less plywood.
To answer your calculus question, we need
$$PS = \dfrac{12 \times SA}{12.8(8)(15)}$$
sheets of plywood ($PS$) and $SA$ is the surface area. We can find
$$\dfrac{d PS}{d SA} = \dfrac{12}{1536}$$
So for every 1 additional square foot of cabin we need $\approx 0.008$ sheets of plywood or it would take $\approx 125$ additional square feet to require $1$ additional sheet of plywood. The same can be done for the foam.