MATH 2301-100 Spring 2022 Calculus I Recitation 7 Week 10

Work in a group of at most 4. Explain to those who do not understand. Ask questions if you do not understand.

  1. (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?
  2. (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?
  3. (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.
    1. How should the wire be cut so that the total area enclosed is a maximum?
    2. How should the wire be cut so that the total area enclosed is a minimum?
  4. (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.

Last modified: Wed Mar 16 15:16:21 UTC 2022