Work in a group of at most 4. Explain to those who do not understand. Ask questions if you do not understand.
(20 points) The radius of a spherical cell is observed to decrease at a rate
of \(2\) units/second when that radius is \(30\) units long. How fast is
the volume of the cell decreasing at that point?
(30 points) A trough is \(10 \mathrm{m}\) long and its ends have the
shape of isosceles triangles that are \(3 \mathrm{m}\) across at the top
and have a height of \(1 \mathrm{m}\). The trough is being filled with
water at a rate of \(12\mathrm{m}^3/\mathrm{min}\). Draw and label a
diagram illustrating this scenario. How fast is the water level rising
when it is \(0.5\mathrm{m}\) deep?
(25 points)
A piece of wire \(10 \, \mathrm{m}\) long is
cut into two pieces. One piece is bent into a square and the
other is bent into an equilateral triangle.
How should the wire be cut so that the total area
enclosed is a maximum?
How should the wire be cut so that the total area
enclosed is a minimum?
(25 points)
A Norman window has the shape of a rectangle surmounted by a
semicircle. (Thus the diameter of the semicircle is equal to the
width of the rectangle.) If the perimeter of the window is \(30\,
\mathrm{ft}\), find the dimensions of the window so that the greatest
possible amount of light is admitted.