Finance work Task Question A client would like his rectangular courtyard of 200 square meters to be designed. The client prefers a large lawn in the middle of the courtyard. He also wants the a paved and tiled border are all round the edge of the lawn. The tiled path is to be 1 meter wide on the east and west, 2 meters wide on the north side and 1.5 meters on the south side as shown below. 2 m 20 ft 1m 1m 1.5 m 10 ft Based on the narrative, what should the length and the width of the rectangular courtyard be, so that the lawn area in the middle of the courtyard is the largest possible?
Answer: since the area of the courtyard is 200 square meters, it cannot be adjusted. Therefore, based on the formula for finding the area of a rectangular field (length*width), the length and the width of the courtyard can vary between 40 and 5, 50 and 4, 100 and 2, 25 and 8, and 16 and 12.5 respectively. If the generated pairs of length and width are multiplied, the resultant area is 200 square meters.
In order to give the courtyard a conspicuous rectangular appearance, the chosen length and width from the above range is 20 meters and 10 meters. The measurements will give the largest area of the lawn in the middle of the courtyard. Based on that, the length and width of the lawn will be (20-3.5) = 16.5 meters and (10-2) = 8 meters. Therefore, the area of the lawn = (L*W) = (16.5 *8) = 132 meters, which is also the largest surface area of the lawn based on the client’s perimeter preference.
Question 2 A mobile phone company has two towers which are 20 feet apart. Tower one is 6 feet high and tower two is 15 feet high. The two towers need to be connected be connected to each other via an expensive cable, which should be attached at the top of the towers. The cable is then connected to the junction box, which is half the distance between the two towers. The cost of the cable connecting the 6 feet tower and the junction box is £ 0.5 million per meter while that connecting the 15 feet tower to the junction box is £ 0.75 million per meter as shown below.
0.75 m 6 ft 0.5 m 15 ft 20 ft (i): To find the length of both the cables, Pythagorean theorem must be applied. Based on the measurement of the 6 feet tower, the high is 6 ft, the base is 10 feet. To find the hypotenuse, which is the length of the cable, Pythagoras theorem is applied. The theory goes, Height^2 + base^2 = Hypotenuse^2.
Therefore, 6^2 + 10^2 = Hypotenuse^2. 36 + 100 = hypotenuse^2. 136 = Hypotenuse^2. Hypotenuse = the square root of 136 = 11.6619 feet (3.5545 meters). On the other hand, the hypotenuse of the 15 feet tower = 15^2 + 10^2 = hypotenuse^2. 225 + 100 = hypotenuse^2. 325 = hypotenuse^2. Hypotenuse = the square root of 325 = 18.0278 feet (5.4949 meters). Based on that, the cost of the 6 feet tower cable = (3.5545*0.5) = £ 1.7773 million, while the cost of 15 feet tower cable = (5.4949*0.75) = £ 4.1212 million.
Therefore, the total cost of the cables = (1.7773+4.1212) = £ 5.8985 million (Right Triangles: The Good Old Pythagorean Theorem n. d.). (ii): in order to determine the location of the junction box to reduce the cost of the cable is determined using trial and error method. The following are the tried location together with the cable costs for both the 6 feet and 15 feet towers. 6 feet tower Height 6 ft 6 ft 6 ft 6 ft Base 2.5 ft 5 ft 7.5 ft 10 ft Hypotenuse (ft) 6.5 ft 7.81 ft 9.6 ft 11.66 ft Hypotenuse (m) 1.9812 m 2.3805 m 2.9261 m 3.5545 m Cable cost (£ m) 0.9906 1.1903 1.4631 1.7773 15 feet tower Height 15 ft 15 ft 15 ft 15 ft Base 2.5 ft 5 ft 7.5 ft 10 ft Hypotenuse (ft) 15.21 ft 15.81ft 16.77 ft 18.0278 ft Hypotenuse (m) 4.636 m 4.819 m 5.111 m 5.4949 m Cable cost (£ m) 3.477 3.614 3.833 4.1212 From the above two tables, the colors (red, yellow, blue and violet represent the possible mixture of the cable lengths.
That is, the red calor under the 6 feet tower table is matched with a similar color under the 15 feet tower. Therefore, the possible costs are £ 4.8236 million for the costs in red color, £ 4.807 million for the costs in yellow color, £ 4.9401 million for the costs in blue color, and £ 5.8985 million for the costs in violet.
Therefore, based on the cost analysis, the minimum total cost for the cables is £ 4.807 million represented by the yellow color. Consequently, the junction box should be placed 5 feet away from the 6 feet tower (15 feet away from the 15 feet tower) (Right Triangles: The Good Old Pythagorean Theorem n. d.). List of References Right Triangles: The Good Old Pythagorean Theorem n. d., Viewed 2 February 2015, http: //www. mathwarehouse. com/geometry/triangles/right-triangle. php