Answer the following:

Please help me on this. The answers to the problems don't have the full solution to its answers. Please give full solution on how did it get the answers. Please give complete solution for all the problem there. Thank you! Geometric Sequences and Series 11.3.1 When light passes through a dust cloud, it decreases in intensity. This decrease can be modeled by a geometric series where each term represents the amount of light lost from the original beam of light entering the cloud. The image to the left shows the dark cloud called Barnard 68 photographed by astronomers at the ESO. Very Large Telescope observatory. The dust cloud is about 500 light years from Earth and about 1 light year across. Answer Key Problem 1 - A dust cloud causes starlight to be diminished by 1% in intensity for each 100 billion kilometers that it travels through the cloud. If the initial starlight has a brightness of B1 = 350 lumens, what is the geometric series that defines its brightness? Answer: B1 = 350 and r = 0.01 so if each term represents a step of 100 billion km in distance, the series is By = 350 (0.99)" n-1 Problem 2 - What are the first 8 terms in this series for the brightness of the light? Answer: Calculate Bn for n = 1, 2, 3, 4, 5, 6, 7, 8 N 1 2 3 5 6 7 8 Bn 350 346 343 340 336 333 330 326 Problem 3 - How far would the light have to penetrate the cloud before it loses 50% of its original intensity? Answer: Find the term number for which Bn = 0.5*350 = 175. Then 175 = 350 (0.99)" n-1 solve for n using logarithms: Log(175) = Log(350) + (n-1) log(0.99) so n-1 = (log(175) - Log(350))/log(0.99) and so n-1 = 68.96 or n = 68. Since the distance between each term is 100 billion km, the penetration distance to half-intensity will be 68 x 100 billion km = 6.8 trillion kilometers. Note: 1 light year = 9.3 trillion km, the distance is just under 1 light year