Applications of derivatives

1103266407

Level: 
C
Our task is to support a square tarp sized $4\,\mathrm{m}\times 4\,\mathrm{m}$ in the centers of its opposite sides to create a hay roof, as shown in the picture. What should be the height $v$ of the pillars to make the shelter space (volume) as large as possible?
$\sqrt2\,\mathrm{m}$
$\frac{\sqrt2}2\,\mathrm{m}$
$2\,\mathrm{m}$
$\sqrt3\,\mathrm{m}$
$\frac{\sqrt3}2\,\mathrm{m}$

1103266406

Level: 
C
The medieval builder has a $5$-ell-long iron belt. His task is to shape the belt into a frame of the Romanesque window (that is the union of a rectangle and a semicircle, see the picture). Find the optimal width $x$ of the window to get as much light coming through the window as possible (i. e. the area of the window should be as large as possible). Express the result rounded in inches ($1\,\mathrm{ell} = 45\,\mathrm{inches}$).
$63$
$140$
$32$
$112$
$83$
$20$

1103266405

Level: 
C
Adam's House ($A$) is located at the distance of $0.9\,\mathrm{km}$ from the road. There is a bus stop ($B$) on this road at the distance of $1.5\,\mathrm{km}$ from the house (see the picture). Adam has overslept and needs to get to the bus stop as quickly as possible. At what distance $x$ from the nearest point $P$ should Adam reach the road knowing that he can move at the speed of $6\,\mathrm{km}/\mathrm{h}$ in rough terrain while being on the road he can move at the speed of $10\,\mathrm{km}/\mathrm{h}$?
$0.675\,\mathrm{km}$
$0.525\,\mathrm{km}$
$0.625\,\mathrm{km}$
$0.575\,\mathrm{km}$

1103266403

Level: 
C
We want to create a rabbit cage in the shape of a rectangle with sides $a$ and $b$. The cage will be divided by parallel walls into four sections with the same area (see the picture). Find the dimensions $a$ and $b$ providing we have $50\,\mathrm{m}$ of fencing wire available and we want the total area to be as large as possible. (Fencing wire will also be used for the walls.)
$a=5\,\mathrm{m}$, $b=12.5\,\mathrm{m}$
$a=4\,\mathrm{m}$, $b=15\,\mathrm{m}$
$a=4.5\,\mathrm{m}$, $b=13.75\,\mathrm{m}$
$a=6.5\,\mathrm{m}$, $b=8.75\,\mathrm{m}$