Section 5.5 – Exponential and Logarithmic Models

A savings account is opened with $20,000 earning 10.5% compounded continuously. a. How many years does it take for the money to double? b. How much is in the account after ten years?

A savings account is opened with $600. At the end of ten years, the account will have $19205. If the account earns interest compounded continuously: a. What is the annual percentage rate of interest? b. How many years did it take for the account to double in value?

Given determine the principal P that must be invested that must be invested at 12% so that $500,000 will be available in 40 years.  

The population P of a city is where t = 0 represents the year 2000. According to this model, when will the population reach 275000?

The population P of a city is where t = 0 represents the year 2000. In 1960, the population was 100,250. Find the value of k, and use this result predict the population in the year 2020.

The number of bacteria N in a culture is modeled by , where t is the time in hours. If N = 300 when t = 5, estimate the time required for the population to double in size.

7. A car that cost $22,000 new has a book value of $13,000 after 2 years. a. Find the straight-line model V = mt + b b. Find the exponential model

7. A car that cost $22,000 new has a book value of $13,000 after 2 years. c. Use a graphing utility to graph the two models in the same viewing window. Which model depreciates faster in the first two years? d. Find the book values of the car after 1 year and after 3 years using each model.

8. The sales S (in thousands of units) of a new product after it has been on the market t years are modeled by . 15000 units of the new product were sold the first year. a. Complete the model by solving for k. b. Use the model to estimate the number of units sold after five years.   55

9. The sales S (in thousands of units) of a product after x hundred dollars is spend on advertising are modeled by . When $500 is spend on advertising, 2500 units are sold. a. Complete the model by solving for k. b. Estimate the number of units that will be sold if advertising expenditures are raised to $700. 3315

10. The management at a factory has found that the maximum number of units a worker can produce in a day is 30. The learning curve for the number of units N produced per day after a new employee has worked t days is . After 20 days on the job, a new employee produces 19 units. a. Find the learning curve for this employee (first, find the value of k). b. How many days should pass before this employee is producing 25 units per day? c. Is the employee’s production increasing at a linear rate? Explain your reasoning.

10. The management at a factory has found that the maximum number of units a worker can produce in a day is 30. The learning curve for the number of units N produced per day after a new employee has worked t days is . After 20 days on the job, a new employee produces 19 units. a. Find the learning curve for this employee (first, find the value of k). b. How many days should pass before this employee is producing 25 units per day? c. Is the employee’s production increasing at a linear rate? Explain your reasoning.

On the Richter scale, the magnitude R of an earthquake of intensity I is where is the minimum intensity used for comparison. Find the magnitude R of an earthquake of intensity I if: a. I = 80500000 b. I = 48275000 c. 251,200