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

We can see that at x = 1, both functions have value 1 and there is one more value of x near 3.513 both have approximately same value 12.34 \r\n y = x^2, and y = e^x - 1 Thus the point of intersection are (1, 1), and (3.513, 12.34) The area of collision between x = 0 to x = 1 is given by integral^1_0 (e^x - 1 - x^2) dx = (e^x - 1 - x^3/3)^1_0 1 - 1/3 - e^-1 + 0 = 0.299 The area of collision between x = 1 to x = 3.513 is given by integral^3.513_1 (x^2 - e^x - 1) dx = (x^3/3 - e^x - 1)^3.513_1 3.513^3/3 - e^3.513 - 1 - 1^3/3 + e^1 - 1 2.77628