Astronomers have a mathematical model for the orbital speed of the stars as a function of their distance x from the center of the galaxy: V(x) = 350x/(1+(x^2))^(3/4) where x = 1 for 10,000 light years, x = 2 for 20,000 light years. At a distance of 10,000 light years, the rotational speed is v(1)= 208 km/sec. The radius of the galaxy M-101 is about 90,000 light years. How fast are stars at this radius orbiting the center?

Question

Astronomers have a mathematical model for the orbital speed of the stars as a function of their distance x from the center of the galaxy: V(x) = 350x/(1+(x^2))^(3/4) where x = 1 for 10,000 light years, x = 2 for 20,000 light years. At a distance of 10,000 light years, the rotational speed is v(1)= 208 km/sec. The radius of the galaxy M-101 is about 90,000 light years. How fast are stars at this radius orbiting the center?

Answer 1

Just plug in x=9 and evaluate to get V(9) = 115.6

Answer 2

That's the answer I got, but my teacher said I was wrong.

Answer 3

To find the orbital speed of stars at a distance of 90,000 light years from the center of the galaxy, we need to evaluate the function V(x) at x = 9.

Given: V(x) = 350x/(1+(x^2))^(3/4)

Step 1: Substitute x = 9 into the equation:
V(9) = 350 * 9 / (1 + (9^2))^(3/4)
= 3150 / (1 + 81)^(3/4)
= 3150 / 82^(3/4)

Step 2: Calculate 82^(3/4) by raising 82 to the power of 3/4:
82^(3/4) = 82^(0.75) ≈ 25.16959

Step 3: Substitute the value of 82^(3/4) back into the equation:
V(9) ≈ 3150 / 25.16959
≈ 125.206 km/sec

Therefore, stars at a distance of 90,000 light years from the center of the galaxy are orbiting the center at a speed of approximately 125.206 km/sec.