Im stuck on this question fir a couple of days now. I need clarification on what the answer is. I know the steps, but what is the answer thank you.

Question

The guardians of the galaxy have to travel in the spaceship Milano from Xander, located on the star map at (-4,-8) to Titan, located at (6,12), to help the avenger stop Thanos. Unfortunately, the fastest route, y=2x, will fly them through an elliptical asteroid field, represented by the equation x^2/16+y^2/4=1. They will travel an average of 0.53 miles per hour, but they must slow down to navigate the asteroid field. While in the field, they will average 0.39 miles per hour. How long will it take them to make the trip?

Answer 1

Answer: 46 hours

Method:

1. t_total = d₁ / v₁ + d₂ / v₂ , v₁ = 0.53 mph. , v₂ = 0.39 mph
2. Finding d₁ and d₂

a. first need to find intersection points of the spaceships path and asteroid field:

x^2 / 16+(2x)^2 / 4=1

17x^2 / 16 = 1

x = +- sqrt(16/17) = +- 0.97014

corresponds to points (-0.97014, -1.9403) , (0.97014, 1.9403)

b. distance traveled in asteroid field (d₂) using distance formula

d₂ = sqrt( (0.97014 + 0.97014)² + (1.9403 + 1.9403)² ) = 2.413 miles

c. total distance traveled

d = sqrt( (6 + 4)² + (12 + 8)² ) = 22.36 miles

d. d₁ = d - d₂ = 19.948 miles

1. t_total = 19.948 miles / 0.53mph + 2.413 miles / 0.39 mph = 46 hours