# Exponential and Logarithmic Functions

**Topics:**Richter magnitude scale, Logarithm, Logarithmic scale

**Pages:**6 (1387 words)

**Published:**December 13, 2014

Individual Project Assignment: Version 2A

Show all of your work details for these calculations. Please review this Web site to see how to type mathematics using the keyboard symbols.

IMPORTANT: See Question 1 in Problem 2 below for special IP instructions. This is mandatory.

Problem 1: Photic Zone

Light entering water in a pond, lake, sea, or ocean will be absorbed or scattered by the particles in the water and its intensity, I, will be attenuated by the depth of the water, x, in feet. Marine life in these ponds, lakes, seas, and oceans depend on microscopic plant life that exists in the photic zone. The photic zone is from the surface of the water down to a depth in that particular body of water where only 1% of the surface light remains unabsorbed or not scattered. The equation that models this light intensity is the following:

𝐼 = 𝐼0 𝑒 −𝑘𝑥

In this exponential function, I0 is the intensity of the light at the surface of the water, k is a constant based on the absorbing or scattering materials in that body of water and is usually called the coefficient of extinction, e is the natural number 𝑒 ≅ 2.718282, and I is the light intensity at x feet below the surface of the water.

1. Choose a value of k between 0.025 and 0.095.

2. In a lake, the value of k has been determined to be the value that you chose above, which means that 100k% of the surface light is absorbed for every foot of depth. For example, if you chose 0.062, then 6.2% of the light would be absorbed for every foot of depth. What is the intensity of light at a depth of 10 feet if the surface intensity is I0 = 1,000 foot candles? (Correctly round your answer to one decimal place, and show the intermediate steps in your work.)

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𝐼

3. What is the depth of the photic zone for this lake? (Hint: 𝐼 = 0.01, so 0.01 = 𝑒 −0.062𝑥 .) 0

Solve this equation for x. Correctly round your answer to one decimal place and show the intermediate steps in your work.

Problem 2: Compound Interest

For discrete periods of time (once per year, twice per year, four times per year, 12 times per year, 365 times per year, etc.), the English terms we use to describe these, respectively, are annually, semiannually, quarterly, monthly, daily, etc. The formula for calculating the future amount when 𝑟 𝑛𝑡

interest is compounded at discrete periods of time is 𝐴 = 𝑃 �1 + 𝑛� , where A is the amount

you will have t years after the money is invested, P is the principal (the initial amount of money invested), r is the decimal equivalent of the annual interest rate (divide the interest rate by 100), and n is the number of times the interest is compounded in 1 year. For the compounding continuously situation, the formula is 𝐴 = 𝑃𝑒 (𝑟)(𝑡) , where A is the amount

you will have after t years for principal, P, invested at r decimal equivalent annual interest rate compounded continuously.

Based on the first letter of your last name, choose values from the table below for P dollars and r percent.

If your last name begins with

Choose an investment amount, Choose an interest rate, r,

the letter

P, between

between

A–E

$5,000–$5,700

9%–9.99%

F–I

$5,800–$6,400

8%–8.99%

J–L

$6,500–$7,100

7%–7.99%

M–O

$7,200–$7,800

6%–6.99%

P–R

$7,800–$8,500

5%–5.99%

S–T

$8,600–$9,200

4%–4.99%

U–Z

$9,300–$10,000

3%–3.99%

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Suppose that you invest P dollars at r% annual interest rate. (Correctly round your answers to the nearest whole penny (two decimal places), and show the intermediate steps in all of these calculations for full credit.)

1. Important: By Wednesday night at midnight, submit a Word document containing only your name and your chosen values from the table above for P and r. Submit this in the Unit 5 IP submissions area. This submitted Word document will be used to determine the Last Day of Attendance for government reporting...

References: Historic earthquakes. (2014). Retrieved from the USGS Web site:

http://earthquake.usgs.gov/earthquakes/states/events/1964_03_28.php

Pidwirny, M. (2010). Earthquake. Retrieved from the Encyclopedia of Earth Web site:

http://www.eoearth.org/view/article/151858/

U.S. Census Bureau. (2012). Health and nutrition. Retrieved from

http://www.census.gov/prod/2011pubs/12statab/health.pdf

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