FA20, Signature Assignment, Oceanfront Property-1 (1)

Signature Assignment Ocean Front Property Your parents are contemplating moving to the beach after they retire. They have found three open lots on Bunting Avenue in Fenwick Island, Delaware—just north of Ocean City. In real estate, ocean front properties are often valued based on the amount of ocean frontage (length of the property line that touches the ocean). In the picture above, assume the non-water property lines that do not lie on Bunting Avenue are perpendicular to Bunting Avenue.

Part A: I calculated the ocean frontage for each of the threes lots by using proportions. I started by drawing the diagram on my sheet of paper then labelling all the parts of the problem. In doing so, I noted that the ocean front property line represents the hypothenuse and the line dividing each property represents transversal lines. From there, I created a triangle within each of the lots by drawing parallel lines on the ocean front property. These lines made three right triangles within each of the lots. I also noted that these triangles had the same bottom-left angle, due to being corresponding angles. From there, I conclude that these angles were similar due to the Angle- Angle postulate. The Angle- Angle postulate states that when two angles are in one triangle are congruent to two angles in another triangle, the two triangles are similar. Since all the triangles are congruent to one another, I can use a proportion to calculate the ocean frontage of each lot. I can form my proportion from creating a large triangle with what the given information, in doing I created dotted lines connecting the 23m to the rest of the provided information on the lots, to form the large triangle. Proportions for Ocean Frontage: Lot A: Lot B: Lot C: x 192 = 61 158 x 192 = 51 158 x 192 = 46 158 158 x = 11712 158 x = 9792 158 x = 8832 11712 / 158 9792 / 158 8832 / 158 x = 74.18 m x = 61.97 m x = 55.90 m I set up the proportion to be the length of each lot to be over the total the distance,152 meters, of all three lots together. The proportion is the x value over the hypotheses, 192 meters. Then I crossed multiplied for each proportion and divided to solve for x. The lots compare to one another by Lot A being the biggest in terms of meters, Lot B being in the middle of Lots A and C in terms of meters and Lot C being the smallest in terms of meters. I would assume that Lot A would be the most expensive because it has the most extensive ocean frontage of 74.8 meters. I would expect that Lot C would be the least expensive because it has the smallest ocean frontage of 55.90 meters.

Lot C: Whole: a 2 + b 2 = c 2 a 2 + b 2 = c 2 46 2 + b 2 = 55.90 2 158 2 + b 2 = 192 2 2116 + b 2 = 3124.81 24964 + b 2 = 36864 3124.81 − 2116 = 1008.81 36864 − 24964 = 11600 b 2 = 1008.81 b 2 = 11600 √ b 2 = √ 1008.81 √ b 2 = √ 11600 b = 31.76 m b = 107.70 m≈ 108 m 108 m + 23 m = 131 m as the missing side length To calculate the area of each lot, I used the area for a trapezoid. The lots within the large triangle would be considered a trapezoid because they are a quadrilateral that has only one pair of parallel sides. I used the formula for the area of a trapezoid which is, A = 1 2 h ( b 1 + b 2 ) . Following this, I converted each lots meter into acres. To do this, I used the conversion of 1 acre= 4046.856 square meters and crossed multiple by my area in squared meters. Then divide by the x value to get lot measure into acres.

Lot A: Conversion: h= 61m 1 acre= 4046.856 b 1 = 23 m 1 4046.856 = x 2687.66 b 2 = 23 + 42.12 = 65.12 m 4046.856 x = 2687.66 2687.66 / 4046.856 x = .66 acres A = 1 2 ( 61 )( 23 + 65.12 ) A = 2687.66 m 2 Lot B: Conversion: h = 51 m 1 acre= 4046.856 b 1 = 23 + 42.12 = 65.12 m 1 4046.856 = x 4318.72 b 2 = 23 + 42.12 + 35.20 = 100.32 m 4046.856 x = 4218.72 A = 1 2 ( 51 )( 65.12 + 100.32 ) 4218.72 / 4046.856 A = 4218.72 m 2 x = 1.04 acres Lot C: Conversion: h = 46 m 1 acre= 4046.856 b 1 = 23 + 42.12 + 35.20 = 100.32 m 1 4046.856 = x 5345.20 b 2 = 23 + 42.12 + 35.20 + 31.76 = 132.08 m 4046.856 x = 5345.20 A = 1 2 ( 46 )( 100.32 + 132.08 ) 5345.20 / 4046.856 A = 5345.20 m 2 x = 1.32 acres I can conclude after these calculations that Lot C has the most acreage at 1.32 acres but the least amount of ocean frontage at 31.76 meters. I found out that Lot B was the middleman between Lot’s A and C at having a modest acreage of 1.04 and ocean frontage of 61.97 meters. Making Lot B about on average .33 away from the other lots in terms acreage and an average of 9.14 meters away from ocean frontage. I also found that Lot A has the least amount of acreage at .66, but the most amount of ocean orientation at 74.13 meters. The property of Lot A would still be the most expensive because oceanfront properties are based on the amount of ocean frontage and not acres. The least costly property would still be Lot C even though it has the most acreage.

Hey Dad, I'm just writing you back to say I finally have completed those calculations you wanted me to look into. First things first I have to say I love the idea of you retiring somewhere that as a beachfront property. I know I will, for sure, visit you as often as I can! Bunting Avenue in Fenwick Island, Delaware is such a nice neighborhood as well, and I looked up it's rating on Zillow. I'm so writing to you give not only my calculations, but my impute as well because you know I'll often be visiting. So, a property that is good for me will be right for you as well. The property I think that is the best for you would be Lot B. It has a bit both in terms of ocean frontage and acreage. Lot B has an ocean frontage of 61.97 meters and acreage of 1.04 acres. Lot B for you would make the most sense because while I know you love to hang at the beach, you also want to have a pool, so Lot B would allow you do have both. It is also at a sensible price. Lot A while it does have the most ocean frontage at,74.18, meters it is the most expensive, I know you don't want to all your saving in one go. Lot A also has the least amount of acreage at, .66 acres so wouldn't have the room to install a pool. Lot B is excellent as well because you can build the further away from the main road of, Bunting Avenue. Lot B is also a cost-efficient option that fits all your real estate wants and needs. Lot C wouldn't be a right choice either because it has the least amount of ocean frontage at 55.90 meters, meaning its not very wide. The pool you so desire lengthwise would be short, so that's reason enough not to purchase. In all Dad, I think your forever home should be Lot B. Lot B is the best of both worlds, you don't have to choose between ocean frontage or acreage because of Lot B as both. On average it is only .33 meters lower on acreage than Lot C. While on average only 9.14 meters smaller than Lot A. Lot B is the best choice, and think of all the fun pool parties as well as beach parties you can have. Reflection: The three objectives from Math 111-01 I choose to reflect one are Develop and use formulas for determining circumference, perimeter, area, and volume, of geometric figures. (GE2, GE3, Use mathematical vocabulary, describe and classify a variety of 2-D and 3-D shapes,

noting their properties and relationships. (GE2, GE3, GE5) and lastly, Convert between units within the metric system. (GE3, GE6). I used “Develop and use formulas for determining circumference, perimeter, area, and volume, of geometric figures. (GE2, GE3)”. I did so when I had to solve for the area of each of the lots stated in the problem. As well as using the Pythagorean theorem, I also used this objective in finding the ocean frontage using a proportion I developed from the given numbers stated in the problems, as well doing some math to check myself and to find missing numbers within the question. Such the height of the large triangle and the base. The next objective I addressed within this assignment was” Use mathematical vocabulary, describe and classify a variety of 2-D and 3-D shapes, noting their properties and relationships. (GE2, GE3, GE5)’, I believe I accomplished this throughout the assignment in describing how I solve for area, how I developed proportions from Angle Angle similarity postulate and identifying the right angles within each of the three lots. The last objective I addressed within this assignment was “Convert between units within the metric system. (GE3, GE6)”. I did this within the assignment when I converted meters squared into acres. I did so by using the known conversion of 1 acre= 4046.856, from there I cross multiples the proportion of 1 4046.856 = x Area of eachlotm 2 , to get the acreage of each lot. All three of these objectives allowed me to complete this assignment. These objectives and the knowledge I have gained over this semester have given me a higher advantage to succeed in future math classes. I will continue to use these objectives and apply these objectives within my college major of Early Childhood Education, and with any geometric problems, I face later down the line. Grading Rubric: