Honors project geometry

maurice and johanna have appreciated the you have provided them and their company pythgo-grass. they have decided to let you consult on a big project.

1, a triangular section of a lawn will be converted to river rock instead of grass. maurice insists that the only way to find a missing side length is to use the law of cosines. johanna exclaims that only the law of sines will be useful. describe a scenario where maurice is correct, a scenario where johanna is correct, and a scenario where both laws are able to be used. use complete sentences and example measurements when necessary.

2, an archway will be constructed over a walkway. a piece of wood will need to be curved to match a parabola. explain to maurice how to find the equation of the parabola given the focal point and the directrix.

3, there are two fruit trees located at (3,0) and (−3, 0) in the backyard plan. maurice wants to use these two fruit trees as the focal points for an elliptical flowerbed. johanna wants to use these two fruit trees as the focal points for some hyperbolic flowerbeds. create the location of two vertices on the y-axis. show your work creating the equations for both the horizontal ellipse and the horizontal hyperbola. include the graph of both equations and the focal points on the same coordinate plane.
4, a pipe needs to run from a water main, tangent to a circular fish pond. on a coordinate plane, construct the circular fishpond, the point to represent the location of the water main connection, and all other pieces needed to construct the tangent pipe. submit your graph. you may do this by hand, using a compass and straight edge, or by using a graphing software program.

5, two pillars have been delivered for the support of a shade structure in the backyard. they are both ten feet tall and the cross sections of each pillar have the same area. explain how you know these pillars have the same volume without knowing whether the pillars are the same shape.