This is a holiday-themed question!
I have a tree with a height of $20 \ \mathrm{ft}$ and a base radius of $7 \ \mathrm{ft}$. I need to order $6\mbox{-}\mathrm{in}$, $4\mbox{-}\mathrm{in}$, and $2\mbox{-}\mathrm{in}$ ornaments.
Assuming the tree is a perfect cone, what's the ratio of ornaments of each size I need such that the bottom third of the tree covered in $6\mbox{-}\mathrm{in}$ ornaments, middle third of the tree covered in $4\mbox{-}\mathrm{in}$ ornaments, and top third of the tree covered in $2\mbox{-}\mathrm{in}$ ornaments maintains a near constant $\%$ of lateral surface area covered in each third?
I've already calculated that
- Bottom third $= \frac{35\pi\sqrt{449}}{9} \ \mathrm{ft}^2 \approx 259. \ \mathrm{ft}^2$, which is $\approx56\%$ of the total area,
- Middle third $= \frac{7\pi\sqrt{449}}{3} \ \mathrm{ft}^2 \approx 155. \ \mathrm{ft}^2$, which is $\approx33\%$ of the total area, and
- Top third $= \frac{7\pi\sqrt{449}}{9} \ \mathrm{ft}^2 \approx 51.8 \ \mathrm{ft}^2$, which is $\approx11\%$ of the total area.
Since we're dealing with area, I assume we can treat the ornaments as circles. Thus,
- $6\mbox{-}\mathrm{in}$ ornaments are effectively $3\mbox{-}\mathrm{in}$ radii circles with areas of $9\pi\approx28.2 \ \mathrm{in}^2$ each,
- $4\mbox{-}\mathrm{in}$ ornaments are effectively $2\mbox{-}\mathrm{in}$ radii circles with areas of $4\pi\approx12.6 \ \mathrm{in}^2$ each, and
- $2\mbox{-}\mathrm{in}$ ornaments are effectively $1\mbox{-}\mathrm{in}$ radii circles with areas of $\pi\approx3.14 \ \mathrm{in}^2$ each.
I'm not sure where to go from here.