Question 1196733
**1. Set up the Coordinate System**

* **Origin:** Place the Sun at the origin (0, 0).
* **Venus's Orbit:** Represent Venus's orbit as a circle centered at the Sun with a radius of 67 million miles. 
* **Comet's Initial Position:** The comet starts at the point (-73, 80) million miles.
* **Comet's Path:** The comet travels along a straight line.

**2. Find the Equation of the Comet's Path**

* **Slope of the Comet's Path:**
    * Slope = (Change in y) / (Change in x) = (0 - 80) / (67 - (-73)) = -80 / 140 = -4/7

* **Equation of the Comet's Path (Point-Slope Form):**
    * y - y1 = m(x - x1)
    * y - 80 = (-4/7)(x + 73)
    * y = (-4/7)x - 32.57 + 80
    * y = (-4/7)x + 47.43

**3. Find the Entry Point of the Comet into Venus's Orbit**

* The comet enters Venus's orbit when its distance from the Sun is 67 million miles.
* We need to find the points on the comet's path that are 67 million miles from the Sun.

* **Equation of a Circle (Venus's Orbit):**
    * x^2 + y^2 = 67^2 

* **Substitute the equation of the comet's path into the equation of the circle:**
    * x^2 + (-4/7)x + 47.43)^2 = 67^2
    * x^2 + (16/49)x^2 - (379.44/7)x + 2249.74 = 4489
    * (65/49)x^2 - (379.44/7)x - 2249.26 = 0

* **Solve the quadratic equation for x:**
    * Using the quadratic formula, we get two solutions for x. 
    * One solution will be the entry point, and the other will be the exit point. 

* **Find the corresponding y-coordinates:**
    * Substitute the x-values into the equation of the comet's path to find the y-coordinates.

* **Determine the Entry Point:**
    * The entry point will be the point where the comet first intersects Venus's orbit.

**4. Calculate the Closest Distance to the Sun**

* The closest distance to the Sun will occur at the point on the comet's path that is perpendicular to a line drawn from the Sun to the comet's initial position.

* **Find the equation of the line perpendicular to the comet's path:**
    * Slope of perpendicular line = 7/4 
    * Equation of perpendicular line: y = (7/4)x 

* **Find the intersection point of the perpendicular line and the comet's path:**
    * Solve the system of equations:
        * y = (-4/7)x + 47.43
        * y = (7/4)x

* **Calculate the distance from the Sun to the intersection point:**
    * Use the distance formula: Distance = √(x^2 + y^2)

**5. Calculate the Time Spent in Venus's Orbit**

* **Find the distance traveled within Venus's orbit:**
    * This is the distance between the entry and exit points.

* **Use the formula: Time = Distance / Speed**
    * Time = (Distance within Venus's orbit) / 0.02 million miles/hour

**Note:**

* This problem involves several steps of algebraic calculations and may require the use of a calculator or computer software to solve the equations accurately.

**I recommend using a graphing calculator or a computer program (like GeoGebra or Desmos) to visualize the problem and assist with the calculations.**

I hope this comprehensive approach helps you solve the Venus Orbit problem!