Venus orbits the sun in an approximately circular path. Venus is about 67 million miles from the sun. A comet is located 80 million miles north and 73 million miles west of the sun. The comet follows a straight-line path and exits Venus's orbit at the east most edge.

a. Find the location where the comet enters Venus's orbit to the nearest tenth?

b. What is the closest distance the comet comes to the sun.

c. If the comet travels at a constant speed of 0.02 million miles per hour, then how long does the comet stay in Venus's orbit

I don't understand any of this to be honest, we have to use secant, perpendicular lines, and/or points of intersections.

let's put the sun at the origin and use x and y in units of 1 million

so the orbit of Venus can be written appr a
x^2 + y^2 = 67^2
then the comet is at (-73,80) which is clearly outside of the orbit of Venus.

the most eastern point on Venus' orbit is (67,0)

it might help to make a sketch here

A line cutting a circle at two points is called a secant
slope of our line = (0-80)/(67+73) = -80/140 = -4/7
equation of line: y = (-4/7)x + b
with (67,0) on it
0 = (-4/7)(67) + b
b = 268/7 ----> y = (-4x + 268)/7

sub that into the equation of the circle:
x^2 + ( (-4x + 268)/7)^2 = 4489
x^2 + (16x^2 - 2144x + 71824)/49 = 4489
times 49
49x^2 + 16x^2 - 2144x + 71824 = 219961
65x^2 - 2144x - 148137 = 0
arrgggh!!!
but it is not so bad, since we know x = 67 works, so
(65x + 2211)(x - 67) = 0
x = -2211/65 or x = 67

a) x = -2211/65 or appr -34.0154, used my calculator to find y = appr 57.7231

(-34.0154 , 57.7231) convert that to suit the question, (multiply by 1,000,000)

verify:
http://www.wolframalpha.com/input/?i=y+%3D+%28-4x+%2B+268%29%2F7+%2C+x%5E2+%2B+y%5E2+%3D+67%5E2

b) closest of (0,0) to y = (-4x + 268)/7
7y = -4x + 268
4x + 7y - 268 = 0
distance to (0,0) = |0+0-268|/√(16+49) = appr 33.2413 million

c) find length of line from (-34.0154,57.7231) to (67,0)
since time = distance/rate .....

carry on

Can you see how this question is the same as

the one that Steve answered for you here ?

http://www.jiskha.com/display.cgi?id=1443993008

Only the numbers have been changed to protect the innocent.

Oh wow, thank you so much I actually got that, just I didn't know how to continue on from there.. but yay now I know how to!!! Thank you so much, I really thought that equation was impossible. Thanks for making it clear, once again(:

a. Well, it seems like the comet is quite the explorer, venturing into Venus's orbit! To find where it enters Venus's orbit, we can use our knowledge of coordinates. The comet is located 80 million miles north and 73 million miles west of the sun. If we draw a straight line from there to the sun, the point where it intersects Venus's orbit will be our answer. So, grab your compass and protractor and let's get calculating!

b. As for the closest distance the comet reaches to the sun, we need to determine the length of the hypotenuse of the right-angled triangle formed by the comet's location. In this case, the distance north is 80 million miles, and the distance west is 73 million miles. To find the hypotenuse, we can use the Pythagorean theorem: a² + b² = c². Prepare for some mathematical magic!

c. Now, let's move on to the comet's speed. It travels at a constant speed of 0.02 million miles per hour. To find out how long it stays in Venus's orbit, we'll need to calculate the distance it travels and then divide it by its speed. So, get your calculator ready!

I hope this humorous breakdown makes it a bit more enjoyable to dive into these calculations. Now, let's embark on this cosmic journey together!

To solve this problem, we can use the concepts of secants, perpendicular lines, and points of intersection. Let's break it down step by step:

a. Find the location where the comet enters Venus's orbit to the nearest tenth:

To find the point where the comet enters Venus's orbit, we need to find the intersection point of the line representing the comet's path and the circle representing Venus's orbit.

Given that the comet is located 80 million miles north and 73 million miles west of the sun, we can represent its location as a point with coordinates (-73, 80) in a Cartesian coordinate system.

Since Venus's orbit is approximately circular and has a radius of 67 million miles, we can represent it as a circle with the center at the origin (0, 0) and a radius of 67 million miles.

To find the intersection point, we need to solve the system of equations formed by the circle equation and the equation of the line representing the comet's path.

The equation of the circle is: x^2 + y^2 = (67 million)^2.

The equation of the line passing through (-73, 80) and the sun (0, 0) can be found using the slope-intercept form: y = mx + c, where m is the slope and c is the y-intercept.

The slope (m) can be calculated as (change in y)/(change in x):
m = (80 - 0) / (-73 - 0) = -80/73.

Therefore, the equation of the line representing the comet's path is: y = (-80/73)x + 80.

Now, to find the intersection points, we substitute this equation into the circle equation:
x^2 + (-80/73)x + 80)^2 = (67 million)^2.

Solving this equation will give us two x-coordinates of the intersection points.

Once we have the x-coordinates, we can substitute them back into the equation of the line to find the corresponding y-coordinates.

b. What is the closest distance the comet comes to the sun:

To find the closest distance the comet comes to the sun, we need to determine the distance between the sun (0, 0) and the point where the comet is closest to it.

Using the coordinates of the point found in part a, we can calculate the distance between the sun and the point using the distance formula:
Distance = sqrt((x - 0)^2 + (y - 0)^2).

c. If the comet travels at a constant speed of 0.02 million miles per hour, then how long does the comet stay in Venus's orbit:

To find the time the comet takes to travel through Venus's orbit, we need to calculate the length of the arc of the circle representing Venus's orbit that the comet travels.

Since the comet travels at a constant speed of 0.02 million miles per hour, we can find the time by dividing the length of the arc by the speed.

The length of the arc can be calculated using the formula: arc length = radius * angle subtended by the arc (in radians).

The angle subtended by the arc can be found by considering the circle as a unit circle and calculating the circumference (2 * pi * radius) divided by the circumference of the entire unit circle (2 * pi * 1).

Once we have the length of the arc, we can divide it by the comet's speed to find the time.

Remember to convert any units as necessary.

By following these steps and applying the relevant formulas, you should be able to find the answers to each part of the problem.