Solve the simultaneous equations.
Given that x and y are positive integers, solve the simultaneous equations:
3^x * 2^y = 72
3^(x-1) * 2^(y+1) = 48
The Riddle of the Rotating Satellite
A communication satellite orbits the Earth in a circular path. At a certain time, its shadow falls on a particular point on the Earth's surface. After 3 hours, the Earth has rotated 45 degrees, and the satellite has also completed a part of its orbit. If the satellite's shadow is now 1000 km away from the original point, calculate the radius of the satellite's orbit, assuming it remains constant and the Earth is a perfect sphere with a radius of 6400 km. Use the formula for the arc length in a circle, S=rθ, where S is the arc length, r is the radius, and θ is the angle in radians.
Given:
ReplyDelete3^x * 2^y = 72
3^(x-1) * 2^(y+1) = 48
First, let's simplify the second equation:
3^(x-1) * 2^(y+1) = 48
=> 3^x / 3 * 2^y * 2 = 48
=> 3^x * 2^y = 48 * 3 / 2
=> 3^x * 2^y = 72
Now, we have two equations:
3^x * 2^y = 144
3^x * 2^y = 144
This means that there are infinite solutions.