Question

I NEED HELP PLEASE, THANKS! :)

Astronomers classify stars according to their brightness by assigning them a stellar “magnitude.” The higher the magnitude the dimmer the star. The dimmest stars visible to the naked eye have stellar magnitudes of 6. The table below shows the relative brightness of different stellar magnitudes.
a. Find an equation that gives the relative brightness in terms of stellar magnitude.
b. Use this equation to find the relative brightness of a star with magnitude 9.
(SHOW WORK)

Answer 1

a. y = 250 (0.4)^x

b. 0.066

Explanation:

The equation can be modeled as a geometric sequence, where the first term is 100 and the common ratio is 0.4.

y = 100 (0.4)^(x − 1)

This can also be simplified to:

y = 250 (0.4)^x

When x = 9:

y = 250 (0.4)^9

y = 0.066

Answer 2

Answer: a) y = 250(0.4)ˣ

b) 0.065536

Explanation:

Create a system of equations in the form y = a(b)ˣ

• where y is the Relative Brightness
• x is the Stellar Magnitude

`Equation\ 1: 100=a(b)^1\\Equation\ 2: 40=a(b)^2\\\\\\\text{Set each equation equal to a to solve for b:}\\Equation\ 1: (100)/(b)=a\qquad Equation\ 2: (40)/(b^2)=a\\\\\\(100)/(b)=(40)/(b^2)\\\\\\100b^2=40b\\\\0.4=b`

Now insert b = 0.4 into one of the equations to solve for "a":

`100=a(0.4)^1\\\\\\(100)/(0.4)=a\\\\\\(1000)/(4)=a\\\\\\250=a`

Now we have our equation for part (a) y = 250(0.4)ˣ

For part (b), replace the x with 9 and solve.

y = 250(0.4)⁹

= 0.065536