MA 093 and Math 117A – Section 4.5 – Exponential Models - Compound Interest& Exponential Decay

  1. A person invests $7000 at 10% interest compounded annually. That is, each year, the amount in the account is ______times the amount of the previous year.Find an equation for the value of the investment after t years.
  1. A person invests $10,000at5% interest compounded annually. That is, each year, the amount in the account is ______times the amount of the previous year.Find an equation for the value of the investment after t years.
  1. On the day you were born, your grandparents set a college fund for you. They deposited $10,000 in an account that paid 8% compounded annually. How much will you have available for college when you turn 18?
  1. The population of a country was 2.5 million people in the year 2000 and since then it has been increasing at a rate of 2% annually. That is, each year, the population is about ______times the previous year’s population. Write an equation for the population of the country (in millions) at t years since the year 2000.
  1. The revenue from music downloads was $1.98 billion in 2007 and has grown by about 86% per year since then. That is, each year the revenue is about ______times the previous year’s revenue.

a)Find an equation for the revenue, R(t) (in billions of dollars) in the year that is t years since 2007.

b)Find R(7) and interpret in context with correct units.

Exponential Decay – Half-Life

  1. A storage tank contains a radioactive element. Let p = f(t) be the amount (in grams) of the element that remains at t years after today. The graph for f is shown below:

/ a)Use the graph to determine the initial amount of radioactive substance in the tank. Use proper units.
b)Use the graph to determine the half life of the element? Use proper units.
c)Use the graph to estimate the amount remaining 70 years from today. Use proper units.
d)Use the graph to estimate when the amount remaining will be 20 grams? Use proper units.

e)Use the graph to read the coordinates of 4 points related to the half-life information starting with the

Y-intercept.Record the coordinates on the table.

X
Y

f)Use algebra and the first two points from the table in part (e) to find the exponential function

y = that fits the data; round to three decimal places.

g)Complete the following: Today, there are ______of radioactive substance in the tank. Every year, the amount remaining is decreasing by subtracting/multiplying (circle one) by ______. Every ______% of the substance remains and ______% decays

h)Use the points from part (e) and the exponential regression feature of the calculator to find the exponential function y = that fits the data; round to 3 decimal places.

i)Use the model equation to find the amount remaining 70 years from today. Compare the answer to your estimate from part (c).

j)Use the model equation and the graphical approach to determine when there will be 20 grams left. Compare the answer to your estimate from part (d).

Graphical Approach

  1. Write the equation
  2. Enter the left hand side of the equation in Y1 of the calculator
  3. Enter the right hand side of the equation in the Y2 of the calculator
  4. Enter appropriate window values.
  5. Press 2nd TRACE [CALC]
  6. Select 5:intersect
  7. Press ENTER three times until you find the point of intersection
  8. Answer the problem

7. A storage tank contains a radioactive element. Let p = f(t) be the amount (in grams) of the element that remains at t years after now. The graph for f is shown below.

(1) Use x-scale = y-scale = 10 and label thetic-marks along the axes.
/ (2) Use x-scale = 20, y-scale = 30 and label the tic-marks along the axes

a)How many grams of the radioactive substance does the tank contain today? Use proper units.
b)Use the graph to determine the half life of the element? Use proper units.
c)Use the graph to read the coordinates of 4 points related to the half-life information starting with the Y-intercept.Record the coordinates on the table.
x
y
/ x
y
d) Use the graph to estimate the amount remaining25 years from today. Use proper units.
e) Use the graph to estimate when the amount remaining will be40 grams? Use proper units
Problem number 7 continued:
f) Use algebra and the first two points from the table in part (c) to find the exponential function that fits the data; round to three decimal places.
g) Use the points from part (c) and the exponential regression feature of the calculator to find the exponential function y = that fits the data; round to 3 decimal places.
h) Interpret the constants a and b from the exponential model following the format shown on part (g) of problem (6)
Problem 7 continued
i)Use the model equation to find the amount remaining 25 years from today. Use proper units.
d)Use the model equation and the graphical approach to determine when there will be 40 grams in the tank.
Graphical Approach
  1. Write the equation
  2. Enter the left hand side of the equation in Y1 of the calculator
  3. Enter the right hand side of the equation in the Y2 of the calculator
  4. Enter appropriate window values.
  5. Press 2nd TRACE [CALC]
  6. Select 5:intersect
  7. Press ENTER three times until you find the point of intersection
  8. Answer the problem
/ Graphical Approach
  1. Write the equation
  2. Enter the left hand side of the equation in Y1 of the calculator
  3. Enter the right hand side of the equation in the Y2 of the calculator
  4. Enter appropriate window values.
  5. Press 2nd TRACE [CALC]
  6. Select 5:intersect
  7. Press ENTER three times until you find the point of intersection
  8. Answer the problem