How do I solve the following word problem : A satellite orbits Earth in a circular path with equation x squared + y squared = 1.44 times 10 to the power of 8. With distances measured in kilometers. Another satellite orbiting in the same plane passes through the point (8000,9800). Is this satellite inside the orbit of the first one? Thank you.
---> Here's what I thought of this question.
1.I should use an equation, so perhaps r^2=(x)^2 + (y)^2.
2.I have to figure out what the radius is though, but I don't know what 10 the power of 8 is, once I do, that will be multiplied by 1.44, and that will be the radius of circle, therefore, any other point from the point of origin to the end point of the circle will be equivalent to what the answer of the first equation is.
As long as I can figure out what 10 to the power of 8 is, I can do the rest on my own. Considering that we are starting from the point of origin, (0,0) it will be 8000^2+9800^2=r^2 Thanks!
the equation of a circle is
x^2 + y^2 = r^2 ---> radius is r
yours:
x^2 + y^2 = 1.44(10^8), so the radius is 1.2(10^4)
you are correct to find
8000^2 + 98000^2 = r^2
r^2 = 9668000000
= 9.668(10^9) , which is clearly > 1.44(10^8)
So the point lies outside the circle of the first satellite.
10^2 = 100
10^3 = 1000
10^4 = 10000 , can you see the pattern?
..
10^8 = 100,000,000
10^9 = 1,000,000,000 , clear bigger than 10^8 by a factor of 10
For the second part i got r=12650.69, am I correct?
Well, it seems like you're on the right track! Let's start by finding the value of 10 to the power of 8. That's actually 100,000,000 - quite a big number, huh?
Now, let's substitute the coordinates (8000,9800) into the equation r^2 = (x)^2 + (y)^2 and solve for r^2.
(8000)^2 + (9800)^2 = r^2
64,000,000 + 96,040,000 = r^2
160,040,000 = r^2
So, the value of r squared for the second satellite is 160,040,000.
Now, let's compare this value to the value of the radius of the first satellite's orbit, which is 1.44 times 10 to the power of 8.
Since 160,040,000 is greater than 1.44 times 10 to the power of 8, the second satellite is actually outside the orbit of the first one.
Well, it looks like the first satellite won't have any company from the second one in its orbit. Those two satellites are keeping their distance, just like some people at social gatherings!
I hope this helps, and keep orbiting those math problems with a smile!
To solve this word problem, you are given the equation of the first satellite's circular path as x^2 + y^2 = 1.44 × 10^8. The next step is to determine if the point (8000, 9800) lies within this circular path.
To do this, you need to substitute the coordinates of the given point into the equation and check if the resulting equation is true. Let's proceed with the calculation:
Using the equation of the circular path:
8000^2 + 9800^2 = r^2
Simplifying this equation:
64,000,000 + 96,040,000 = r^2
160,040,000 = r^2
Taking the square root of both sides to find the radius:
r = √(160,040,000)
Calculating the square root:
r ≈ 4000
Now, we have calculated that the radius of the circular path is approximately 4000 kilometers.
To determine if the point (8000, 9800) is inside the orbit of the first satellite, we need to calculate the distance between the origin (center of the circular path) and the given point. If this distance is less than the radius, the point is inside the orbit; otherwise, it is outside.
Using the distance formula:
distance = √((x2 - x1)^2 + (y2 - y1)^2)
Substituting the values:
distance = √((8000 - 0)^2 + (9800 - 0)^2)
distance = √(8000^2 + 9800^2)
distance ≈ √(64,000,000 + 96,040,000)
distance ≈ √160,040,000
distance ≈ 4000
We have calculated that the distance between the origin and the point (8000, 9800) is also approximately 4000 kilometers. Since this distance is equal to the radius of the circular path, the given point lies on the edge of the orbit of the first satellite, not inside it.
Therefore, the second satellite is not inside the orbit of the first one.
To solve the word problem, you can follow these steps:
1. Start with the given equation: x^2 + y^2 = 1.44 × 10^8. This equation represents the orbit of the first satellite.
2. To find the radius of the orbit, you need to isolate the term with the variable r (radius). Rearrange the equation as follows: r^2 = x^2 + y^2.
3. Now, you need to calculate the radius. Plug in the values of x = 0 and y = 0 (since it represents the point of origin, (0,0)), and solve for r: r^2 = 0^2 + 0^2 = 0. Therefore, the radius is 0.
4. Given that another satellite passes through the point (8000,9800), you need to determine if that point lies inside the orbit of the first satellite.
5. Calculate the distance from the center (origin) to the point (8000,9800) using the distance formula: r^2 = x^2 + y^2. Substituting the values, you get: r^2 = (8000)^2 + (9800)^2.
6. Evaluate the equation to find the value of r^2, which represents the distance from the origin to the point (8000,9800).
7. Compare the value of r^2 from step 3 (which is equal to 0) with the value of r^2 from step 6. If the value of r^2 from step 6 is less than 0, then the point (8000,9800) lies inside the orbit of the first satellite. Otherwise, if the value of r^2 from step 6 is equal to or greater than 0, then the point lies outside the orbit.
By following these steps, you can determine whether the second satellite lies inside the orbit of the first satellite based on the given equation and the distance relationship.