Hello. I am a type of learner that likes to see the outcome of something till the end. Im just confused on this problem overall and i just need the answer thank you so much.

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

I will walk you through some of this...but some of it you will need to do by yourself!

The line y=2x passes through (-4,-8) and (6,12).

Use the distance formula to find D the total distance form (-4,-8) to (6,12)

This path intersects the ellipse at 2 points.

Substitute y=2x into the equation for the ellipse to get the x values for the 2 points=±(4/17)sqrt17).

Use these values to get the y co-ordinates of the points of intersection: ±(8/17)sqrt(17).

Now use the distance formula again to find the distance d that the path is inside the ellipse.

Use d to find the time to cross the ellipse.

Then use D-d to find the time to traverse the distance outside the ellipse.

The sum of those 2 times is your answer.

Answer 2

1. First, find the points where the line y = 2x intersects the ellipse. Do this by substituting 2x in place of y in the equation for the ellipse then solve for x. You'll get ± a square root. If allowed, you can use your calculator to approximate the answer. Round to two decimal places. These are the x-coordinates of the intersection points. To get the y-coordinates, plug the two values of x into the equation y = 2x.
2. Next use the distance formula to compute the distance between the two intersection points. time = distance/velocity, so the time the spaceship spends flying through the ellipse is the distance between the two intercept points divided by 0.39 mph. Call this time1.
3. Third, use the distance formula to compute the distance between (-4,-8) and (6,12). Then subtract the distance that the spacecraft travels in the ellipse (from step 2 above). This is the distance that the space craft travels at 0.53 mph. The time it takes is this distance divided by 0.53 mph. Call this time2.
4. Total time to travel between Xander and Titan is time1 + time2. The units are hours. Post your answer as a comment and I or another tutor will check it for you.