Question 3
An investor has two rectangular plots of land, each measuring $120\,\mathrm m\times100\,\mathrm m$.
(a) On the first plot, the investor wants to build a house with a rectangular footprint measuring $25\,\mathrm m\times35\,\mathrm m$, and nine circular flowerbeds outside the house, each of diameter $5\,\mathrm m$. Prove that wherever the nine flowerbeds are built first, enough space remains for the house.
(b) On the second plot, the investor wants a pond shaped as a convex polygon, such that from any point of the remaining land one can reach the pond's bank by walking no more than $5\,\mathrm m$. Prove that the pond's perimeter is at least $(440-20\sqrt2)\,\mathrm m$.
(a) On the first plot, the investor wants to build a house with a rectangular footprint measuring $25\,\mathrm m\times35\,\mathrm m$, and nine circular flowerbeds outside the house, each of diameter $5\,\mathrm m$. Prove that wherever the nine flowerbeds are built first, enough space remains for the house.
(b) On the second plot, the investor wants a pond shaped as a convex polygon, such that from any point of the remaining land one can reach the pond's bank by walking no more than $5\,\mathrm m$. Prove that the pond's perimeter is at least $(440-20\sqrt2)\,\mathrm m$.