Math 2414 Activity 12(Due by August 7)
1.Four bugs are placed at the four corners of a square with side length a. The bugs crawl counterclockwise at the same speed and each bug crawls directly toward the next bug at all times. They approach the center of the square along spiral paths.
a) Find the polar equation of a bug’s path assuming that the center of the square is the origin.
Notice that each bug’s path is the previous bug’s path rotated by radians, so let’s just find the path of the bug that starts in the first quadrant. Let’s call its path . At the point the slope of the tangent line is given by , but it’s also given by . So solve the DE with initial condition:.
b) What’s ?
c) Find the total distance traveled by a bug as it makes its way towards the origin.
2. a) Attempt the same problem, except that there are three bugs at the corners of an equilateral triangle of side length a centered at the origin with one vertex along the positive x-axis
Notice that each bug’s path is the previous bug’s path rotated by radians, so let’s just find the path of the bug that starts on the x-axis. Let’s call its path . At the point the slope of the tangent line is given by , but it’s also given by. Simplify using the identities: and , and solve the DE with initial condition.
b) Find the total distance traveled by a bug as it makes its way towards the origin.
3. Find the area of the intersection of the two circles and , where a is a positive constant.
4. Find the area inside the circle but outside the circles and .
{No calculus is required or recommended!}
5. A farmer has a fenced circular pasture of radius a and wants to tie a cow to the fence with a rope of length b so as to allow the cow to graze half the pasture. How long should the rope be to accomplish this?
The length of the rope,b, must be longer than a and shorter than , i.e. . To find the area of the grazing region, we can use polar coordinates:
The grazing area .
We want this to equal half the pasture area which is , so we get the equation . If we multiply both sides by and perform the integrations, we arrive at the equation .
a) Verify the previous equation.
If we let , we get the simplified equation , and we’re looking for the solution x, with . Let’s rearrange it into .
Here’s a plot of the left side of the equation with .
b)Approximate the solution by performing the Bisection Method on the interval :
Left(sign) / Midpoint(sign)Estimate / Right(sign) / Maximum Error
1(-) / (+) / (+) / .25
1(-) / (-) / (+) / .125
So this means that the length of the rope, b, should approximately equal a times your best estimate of the solution:
6.Find the area bounded by the loop of the folium of Descartes with equation . Start by converting the equation to polar coordinates and then use the substitution .
7. The line is tangent to the graph of . Find a.
8. Find the vertices of the triangle determined by the three lines , , and .
9. Find the area of the triangle determined by the three lines , , and .
10. Consider the polar equation .
a) The polar equation is equivalent to the parametric equations:
, .
Show that the entire graph of lies within the vertical strip .
b) Show that the vertical line is a vertical asymptote for the graph of .
{Hint: Consider and , as well as and .}
11. The length of paper on a tightly wound roll needs to be determined. The roll of paper has an inner radius of 3 cm, an outer radius of 5 cm, and the thickness of the paper is 1mm.
A polar function whose graph models the coiled paper is given by a spiral whose radius increases by .1 cm for each rotation of radians. So .
Determine the length of this polar curve to estimate the length of the coiled paper.
12. The formula for the area of an ellipse, , can be derived from the parametric equations of the ellipse:. The area is given by . Or in other words, the area of an ellipse is the product of the lengths of the major and minor axes and . Interpret the integral as the area of a certain ellipse containing the origin in order to find its value.
{Hint: If is the polar integral for the area inside an ellipse containing the origin, then . This gives you the polar equation of the ellipse, so use it to determine the lengths of the major and minor axes.}
13. Find as many of the five points of intersection of the two curves and as you can. You can easily see four of the intersection points from the graph. To find the fifth intersection point, zoom-in at the origin.
14. Find the length of the curve .
{Hint: .}
15. Find the length of the curve . Approximate, if necessary.
16. Suppose that the polar curve with has continuous and length L. Find the length of the polar curve with for any constant c.
{Hint: Be careful, c could be negative.}
17. Suppose that the polar curve with and continuous encloses a region of area A. Find the area of the region enclosed by the polar curve with for any constant c.
{Hint: Be careful, c could be negative.}
18. a) Show that the polar equation , for a and b real numbers not both zero, describes a circle.
{Hint: Multiply both sides by r and use , , and .}
b) Find formulas for the center and radius of such a circle.
c) Find a formula for the area enclosed by such a circle.
19. a) For two points with rectangular coordinates and , the formula for the distance between the points is . Derive the distance formula for two points with polar coordinates and .
{Hint: Use the rectangular distance formula, the fact that and , and the identity }
b) Use the distance formula in polar coordinates to find the distance between the two points with polar coordinates and .
20. The parabola divides the area of the circle into two parts. Find the area of the smaller part.
{Hint: The polar equation of the circle is , and the polar equation of the parabola is .}
21. Find the area inside the lemniscate , but outside the lemniscate .
22. Find the volume generated by revolving the area enclosed by the cardioid about the x-axis.
23. The graph of the polar coordinate equation is called a conchoid of Nicomedes. The graph is unbounded, but it contains a closed loop. Find the area of the region inside the loop.