Question

9. Part of the trajectories of two communication satellites are represented by the parabolic and exponential equations: Ime y x2 and y= e*-1 It is believed that there are two regions where the trajectories may overlap resulting in a collision danger. a) Plot the two trajectories and determine the values of x and y where they intersect. b) Use definite integration to determine the area of both possible collision zones.

Answer 1

a y exp(x-1) y=x^2 Chart Title 0.367879441 25 0.2 0,04 0.449328964 0.4 0.16 0.548811636 0.6 0,36 0.670320046 20 0.818730753 0.8 0.64 1 1 1 1.2 1.44 1.221402758 15 1.4 1.96 1.491824698 1.6 2.56 1.8221188 1.8 3.24 2,225540928 10 2 4 2.718281828 2.2 4.84 3.320116923 у %3D х* 2.4 4.055199967 5.76 y = e*-1 2.6 6.76 4.953032424 2.8 7.84 6.049647464 9 7.389056099 0 - 3.2 9.025013499 0 0.5 1 1.5 2 25 3 3.5 4 5 10.24 3.4 11.56 11.02317638 x^2 —уех p(х-1) 3.6 12.96 13.46373804 3.8 14.44 16.44464677 We can see that at x=1, both functions have value 1 and there is one 4 16 20.08553692 more value of x near 3.513 both have approximately same value 12.34

y x2 and y = e*-1 (1, 1), and (3.513, 12.34) Thus the point of intersection are b) The area of collision between x = 0 to x= 1 is given by 1 1 r-1 3 1 e-10 0.299 3 = 1 1 to x 3.513 is given by The area of collision between x = 3.513 3.513 3 e-1 3 (x2-e*-1)dx 1 1 3.5133 13 e1-1 3 e3.513-1 3 -2.77628