Question

2. The city plans to use a newly acquired plot of land to create a multi-functional public space. A rectangular grassy field will be laid down, with two semi-circular concrete courtyards bordering the east and west sides of the green. A running track will be placed around the property (courtyards and field combined); see the figure provided below. If the field alone is to cover 600 square feet, what radius should be used for the concrete courtyards to ensure that the running track is as long as possible? TRACK RUNNING COURTYARD #1 FIELD COURTYARD #2 * Figure not to scale

Answer 1

© , 9 y Let length of field be x and too width bey Giren ny 2. - Goo feet? Than sudius of coursty and = / Я +27 Length of Running Track = (IT (Y₂) + x) (2) Length of Track = ття say length of Track flasy) = TTY +2* Now to max flay) = TTY +2x sub х = Goo using Lagringe multiplier 2 С ху) ту+2х + (Сое — ху) 2 - 2 = 0 27 эх - 뜷 - => П 7 әЯ T = » 22 Cred 2 o e) у = боо 륭 Since - 1 - 2 = T- 2

09 19 - Goo ту 2 боо 2 2 12оо 9 T Я 200 3 Я я 2O Formax, с То зл 8 9 1 / 4 feet 9/, radius -