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.

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.

**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.

Elliptical Flowerbed

3=a^2-b^2

3=a^2-8^2

3=a^2-64

67=a^2

sqrt(67)=sqrt(a^2)

8.19=a

**Question 4**

9.05 Honors Activity

For example, the triangular part of a lawn is named ABC and the sides are abc as the opposite of the angles with the similar letters. The missing length here is side b. Maurice used the Law of Cosines to get this. For him to do this, he would need two sides and an angle between them that is given to solve for the missing length which is sides a and c and angle B. Johanna used the Law of Sines. To use this she needs two angles and an opposite side that should be given to find the missing length which is angle B and C and side c. Since both laws were used the remaining given, the use of either would result in similar measurements.

Question 2

To solve the equation of the parabola, you would need to use the distance formula. You would also use this formula for the directrix. The X could be eliminated and the remaining part of the formula would be used. After that you substitute the Y for the given measurement of the directrix. You would then combine the two unfinished formulas and solve the equation. f(x) would be the unit of measurement for the parabola.

By: Markelle Maddock

x^2/a^2+y^2/b^2=1

x^2/18.9^2+y^2/8^2

x^2/67.0761+y^2/64=1

Hyperbolic Flowerbed

a^2+b^2=c^2

2^2+b^2=3^2

4+b^2=9

5=b^2

sqrt(5)=sqrt(b^2)

2.24=b

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

x^2/2^2-y^2/2.24^2=1

x^2/4-y^2/5.0176=1

**Question 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

Based from Cavallieri's Principle, it is said that when two solids have the same height and their cross sections reveal the same area, they both have the same volume.

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 GeoGebra.