A trajectory of a spaceship traveling from Earth to Mars can be expressed as the equation where the sun is the origin and is expressed in million mi. Earth is 151.1 million mi. from the sun, and mars is 211.7 million mi. from the sun.
a. What type of polar graph does this equation represent?
b. How many radians around the sun does the spaceship travel?
The trajectory equation is r=28.93 theta
(a) this is a spiral
(b) r(5.223) = 151.1
r(7.318) = 211.7
So θ changed by 2.095
To answer these questions, we need the equation mentioned. However, it seems that part of the equation is missing from your question. Please provide the complete equation in order to proceed with the analysis.
To determine the type of polar graph represented by the given equation and find the number of radians the spaceship travels, we need to convert the given information into polar coordinates and evaluate the equation.
a. The equation representing the trajectory of a spaceship traveling from Earth to Mars can be expressed as r = a + bθ, where r represents the distance from the origin (the sun), θ represents the angle around the origin, and a and b are constants.
Let's convert the given distances of Earth and Mars into polar coordinates:
Distance of Earth from the sun (151.1 million mi.):
Earth's distance can be expressed as r = 151.1 million mi.
Distance of Mars from the sun (211.7 million mi.):
Mars's distance can be expressed as r = 211.7 million mi.
Since the equation of the trajectory is not given, we assume it to be r = a + bθ, where a and b are constants to be determined.
Comparing the given distances with the form of the equation, we get the following equations:
151.1 = a + bθ1, where θ1 represents the angle at which Earth is located.
211.7 = a + bθ2, where θ2 represents the angle at which Mars is located.
Since the origin is the sun (r = 0), we can express the angles θ1 and θ2 as θ1 = 0 radians and θ2 = π radians, respectively.
Now, we have two equations with two unknowns (a and b):
a + b(0) = 151.1
a + b(π) = 211.7
Simplifying the equations:
a = 151.1
a + bπ = 211.7
Substituting the value of a into the second equation:
151.1 + bπ = 211.7
Solving for b:
bπ = 211.7 - 151.1
b = (211.7 - 151.1) / π
b ≈ 60.6 / π
Thus, the equation representing the trajectory is r ≈ 151.1 + (60.6 / π)θ.
To determine the type of polar graph represented by this equation, we need to look at the coefficient of θ. Since the coefficient (60.6 / π) is positive, the graph will either be a spiral with the distance from the origin increasing as θ increases or a cardioid.
b. To find the number of radians the spaceship travels around the sun, we need to find the range of values for θ between Earth and Mars.
Since Earth is located at θ = 0 radians and Mars is located at θ = π radians, the spaceship travels θ = π radians around the sun.
In summary:
a. The equation representing the trajectory is r ≈ 151.1 + (60.6 / π)θ, which can be either a spiral or a cardioid.
b. The spaceship travels θ = π radians around the sun.