**Honors Extension Activity**

Problem 1

Question:

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.

Problem 2

Question:

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.

Problem 3

Question:

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.

Problem 4

Answer:

Let’s say that triangle ABC represents the triangular section of lawn that will be converted to river rock. The opposite of angle A is a, the opposite of angle B is b, and the opposite of angle C is c. We are solving for line c. Maurice wants to use the law of Cosines to solve for the missing angle of triangle ABC. If they had the measurements for lines a and b only, then you would only be able to solve for line c with the law of cosines. Johanna wants to use the law of Sines to solve for line c. If only the measurements for lines a, b, and angle A were known, then you could use law of sines to solve for line c. As long as 2 sides and an opposite angle measure are know, both the law of Sines and the law of Cosines can be used to determine the missing side length c.

Answer:

First, draw a perpendicular line from the directrix passing through the focus, this will be the line of symmetry. First, use the distance formula, see figure 1, your 2 points being the focus, and (x,y). This formula represents the distance from the focus, to any point on your parabola. This equation is equal to the square root of (the directix - y) squared. See figure 2. Now simplify your equation.

Figure 1

Figure 2

Answer:

Maurice wants to create a set of elliptical flower beds. To do this, he first plots the location of the two fruit trees on his graph. Maurice has to use the equation a^2-b^2=c^2. We know that c=3, and because we need 1 more number to solve for b, I made a=6. 6^2-b^2=3^2. 36-b^2=9. b^2=27. b=5.196

Next, to create the equation, we substitute what we know into the equation x^2/a^2 + y^2/b^2=1 and get x^2/36 + y^2/27=1. Johanna wants to create some hyperbolic flower beds. We already know that c=3 so this time I decided a=1. 3^2=1^2+b^2. 9=1+b^2. 8=b^2. b=2.828 Next, to create the equation, we substitute what we know to the equation x^2/a^2 - y^2/b^2 = 1. x^2/1^2 - y^2/2.828^2 = 1.

Problem 5

Question:

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.

Answer:

I know these pillars have the same volume because Cavalieri's Principle states that any two solids with the same height and if it is proven that the two solids have the same area in their cross sections, then they have the same volume.