**9.05 Honors Extension Activity**

Question 1

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.

For the law of cosines to be used, two sides and the angle between them must be given to find the third side or all 3 sides must be given to find an angle. To use the law of sines, two sides and an opposite angle must be given to find another angle or two angles and an opposite must be given to find another side. In order for Johanna to be correct, the group would have to have the triangular section of a lawn that contains measurements for side a and b and the opposite sides. In Maurice's favor there must be a triangular section of the lawn that contains measurements for sides a and b and the angle between them. As long as two sides and an opposite angle measure are known, both of these lawns can be used.

Question 2

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.

You draw a line of symmetry through the focus then you use the distance formula to simplify your focus points. This gives you the distance between your focus point and any point of the parabola. This will equal (y2-y1) ^ 2. Y1 equals the directrix, then you simplify.

Question 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.

Maurice wants elliptical flower beds.

He has to use the equation --> a^2-b^2=c^2.

C= 3 and I made a equal to 6.

6^2 - b^2 = 3^2. 36-b^2 = 9. b^2 = 27 b=5.196

--> x^2/a^2 + y^2/b^2 --> x^2/36 + y^2/27 = 1

Johanna wants hyperbolic flower beds

a will be 2

2^2+b^2=3^2 --> 4 + b^2 = 9 --> b^2 = 5 --> b = 2.236

x^2/a^2-y^2/b^2 = 1 --> x^2/2^2 - y^2 / 2.236^2 = 1

Question 3

Question 3

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.

These two pillars have the same volume because Cavalierie's Principle states that any two solids with the same height and if it's also proven that they have the same area in their cross sections then their volume is the same.

Question 4

Question 5