Question

1. Mr. Wahnon’s Vintage Sound Systems Emporium has experienced a decrease in its sales since opening in 2010. Since then, sales of 8-track players have been decreasing at a steady rate of 5.1% per year. If the Emporium sold 390 8-track players in 2010, write an exponential function to determine the number of players sold in 2013 and give the amount to the nearest whole unit. ***2 separate answers****
2. Ms. Holder was playing with the science equipment last night and accidentally spilled a solution containing Neptunium-229. This isotope has a half life of 4 minutes. If there were 500 grams in the solution originally at 7:45 am, how much remains at 8:15 am? Round your answer to the nearest 0.01 grams.

3. Write an equation to model the situation: Mr. Canonaco starts a bank account in Florida with his paycheck ($2475.76) at Dewey, Cheetum, & Howe Credit Union. It will collect 6.25% in annual interest over a period of x years.

Answer 1

1. Since the sales of 8-track players are decreasing by 5.1% per year, we can say that the sales after a year are 94.9% of the sales of the previous year. For succeeding years, we just multiply the percentage a number of times corresponding to the number of years. Forming an exponential function for this we'll get:

$f(x) = 390*0.949^{x}$ where x is the number of years and f(x) is the number of sales.

To find the number of sales in 2013 we just plug in 3 for the value of x:

$f(3) = 390*0.949^{3}=333$

ANSWER: Emporium sold 333 8-track players in 2013.

2. The half life of Neptunium-229 is modeled by $A=500* \frac{1}{2} ^{ \frac{t}{4} }$ where A is the final amount and t is the time that has passed in minutes. This just means that the original amount will be multiplied by one half for every 4 minutes.

30 minutes has passed from 7:45 am to 8:15 am. Therefore, we just substitute 30 as the value of t to find the amount of Neptunium remaining by 8:15.

$A=500* \frac{1}{2} ^{ \frac{30}{4} }=2.76$

ANSWER: 2.76 grams of Neptunium-229 are remaining at 8:15 am.

3. To find how much annual interest is collected over x years, we just do the same modelling as the previous items. Knowing that the initial amount that Mr. Canonaco deposited into his bank account and that the Dewey, Cheetum, & Howe Credit Union collects 6.25% annual interest per year, the equation will be as follows:

$f(x)=2475.76*0.0625^{x}$ where x is the number of years and f(x) is the annual interest collected.