jmc

Question

Five points A, B, C, D, and O lie on a flat field. A is directly north of O, B is directly west of O, C is directly south of O, and D is directly east of O. The distance between C and D is 140 m. A hot-air balloon is positioned in the air at H directly above O. The balloon is held in place by four ropes HA, HB, HC, and HD. Rope HC has length 150 m and rope HD has length 130 m. How high is the balloon above the field (that is, the length of OH)? FIGURE

Accepted Answer

Let OC=c, OD=d and OH=h. FIGURE Note that OH is perpendicular to the field, so OH is perpendicular to OC and to OD. Also, since OD points east and OC points south, then OD is perpendicular to OC. Since HC=150, we have h^2+c^2=150^2 by the Pythagorean Theorem. Since HD=130, we have h^2+d^2=130^2. Since CD=140, we have c^2+d^2 = 140^2.. Adding the first two equations, we obtain 2h^2+c^2+d^2=150^2+130^2. Since c^2+d^2=140^2, we have align* 2h^2 + 140^2 &= 150^2+130^2 2h^2 & = 150^2 + 130^2 - 140^2 2h^2 & = 19800 h^2 & = 9900 h & = 9900=3011 align* Therefore, the height of the balloon above the field is 3011 meters.