Logarithms; Sequences & Series; Algebra Problems
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Q # 4.
The loudness of sound is based on intensity level measured in decibels using a logarithmic scale and is relative to (a ratio of) the weakest sound the ear can hear?
Research how sound is measured. Answer the following:
? The formula for measuring sound.
? Pick a specific sound, give the decibels of the sound, and explain what this measurement means.
Please be sure to cite all references using APA style
Q # 5.
Using the index of a sequence as the domain and the value of the sequence as the range, is a sequence a function?
Include the following in your answer:
? Which one of the basic functions (linear, quadratic, rational, or exponential) is related to the arithmetic sequence?
? Which one of the basic functions (linear, quadratic, rational, or exponential) is related to the geometric sequence?
? Give at least two real-life examples of a sequences or series. One example should be arithmetic, and the second should be geometric. Explain how these examples would affect you personally.
Questions 1 - 2: Solve equations involving square and cube roots; determine if an identity is true
Questions 3 - 5: Graph the function, identify the graph of the function (line, parabola, hyperbola, or exponential), explain your choice, and give the domain and range as shown in the graph, and also the domain and range of the entire function; give the y-value for the function; solve for all values of x that satisfies the equation.
Question 6: A right triangle is a triangle with one angle measuring 90°. In a right triangle, the sides are related by Pythagorean Theorem, where c is the hypotenuse (the side opposite the 90° angle). Find the hypotenuse when the other 2 sides' measurements are 3 feet and 4 feet.
Question 7: Suppose you travel north for 35 kilometers then travel east 65 kilometers. How far are you from your starting point? North and east can be considered the directions of the y- and x-axis respectively.
Question 8: The volume of a cube is given by V = s3. Find the length of a side of a cube if the Volume is 729 cm3.
Question 1: An open-top box is to be constructed from a 6 by 8 foot rectangular cardboard by cutting out equal squares at each corner and the folding up the flaps. Let x denote the length of each side of the square to be cut out.
Question 2: The volume of a cylinder (think about the volume of a can) is given by V = pir2h where r is the radius of the cylinder and h is the height of the cylinder. Suppose the volume of the can is 100 cubic centimeters.
Question 3: The formula for calculating the amount of money returned for an initial deposit money into a bank account or CD (Certificate of Deposit) is given by
A is the amount of returned.
P is the principal amount initially deposited.
r is the annual interest rate (expressed as a decimal).
n is the compound period.
t is the number of years.
Suppose you deposit $10,000 for 2 years at a rate of 10%.
Calculate the return (A) if the bank compounds annually (n = 1). Round your answer to the hundredth's place.
Question 4: For a fixed rate, a fixed principal amount, and a fixed compounding cycle, the return is an exponential function of time. Using the formula, , let r = 10%, P = 1, and n = 1 and give the coordinates (t, A) for the points where t = 0, 1, 2, 3, 4. Round the A value to the tenth's place.
Question 5: Using a calculator, find log 10000 where log means log to the base of 10.
Question 1: Use the arithmetic sequence of numbers 1, 3, 5, 7, 9,...to find the following ....
Question 2: Use the geometric sequence of numbers 1, 2, 4, 8,...to find the following ....
Question 3: Use the geometric sequence of numbers 1, 1/2, 1/4, 1/8,...to find the following ....
Question 4: CLASSIC PROBLEM - A traveling salesman (selling shoes) stops at a farm in the Midwest. Before he could knock on the door, he noticed an old truck on fire. He rushed over and pulled a young lady out of the flaming truck. Farmer Brown came out and gratefully thanked the traveling salesman for saving his daughter's life. Mr. Brown insisted on giving the man an award for his heroism.
So, the salesman said, "If you insist, I do not want much. Get your checkerboard and place one penny on the first square. Then place two pennies on the next square. Then place four pennies on the third square. Continue this until all 64 squares are covered with pennies." As he'd been saving pennies for over 25 years, Mr. Brown did not consider this much of an award, but soon realized he made a miscalculation on the amount of money involved.
How much money expressed in dollars would Mr. Brown have to put on the 32nd square?
How much money expressed in dollars would the traveling salesman receive in total if the checkerboard only had 32 squares?
Calculate the amount of money necessary to fill the whole checkerboard (64 squares). How money expressed in dollars would the farmer need to give the salesman?
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Solution Summary
There are several questions in this problem set:
-- a discussion of the measure of sound (and how it relates to logarithms)
-- definitions, examples, and problems involving arithmetic and geometric sequences and series
-- examples of solving equations involving square and cube roots
-- graphing functions and determining the type of function and the domain and range
-- solving word problems involving right triangles and volumes of cubes and cylinders
-- problems involving compound interest
Solution Preview
See the attached two files.
Q # 4.
The loudness of sound is based on intensity level measured in decibels using a logarithmic scale and is relative to (a ratio of) the weakest sound the ear can hear?
Research how sound is measured. Answer the following:
? The formula for measuring sound.
? Pick a specific sound, give the decibels of the sound, and explain what this measurement means.
Please be sure to cite all references using APA style
Answer:
A decibel is a unit used to express the amount of a change in level of power, voltage, current, or sound. Decibels are measured using a logarithmic scale of base 10.
The difference in intensity of two sounds can be calculated using the following formula:
Decibels (dB) = 10 log(P2/P1)
where P1 and P2 are the intensities of the two sounds you are comparing. Usually, P1 is a fixed value used as a reference, most often zero decibels, which is defined to be the threshold of hearing, 0.0002 μbar ("μbar" means microbars; a bar is the "normal" pressure of air).
A sound that is 10 times as loud as something of zero decibels is 10 dB, a sound that is 100 times as loud is 20 dB, a sound that is 1000 times as loud is 30 dB, and so on.
The following websites clearly explain the concept of decibels and give examples of the decibel levels of certain sounds: http://www.phys.unsw.edu.au/~jw/dB.html, https://ewhdbks.mugu.navy.mil/decibel.htm, and http://en.wikipedia.org/wiki/Decibel.
Q # 5.
Using the index of a sequence as the domain and the value of the sequence as the range, is a sequence a function?
Include the following in your answer:
? Which one of the basic functions (linear, quadratic, rational, or exponential) is related to the arithmetic sequence?
? Which one of the basic functions (linear, quadratic, rational, or exponential) is related to the geometric sequence?
? Give at least two real-life examples of a sequences or series. One example should be arithmetic, and the second should be geometric. Explain how these examples would affect you personally.
Answer:
An arithmetic sequence is a sequence created by adding a constant to the preceding term. An example is the odd numbers: 1, 3, 5, 7, ... which are formed by adding 2 to each term, starting with 1.
The general equation for an arithmetic sequence is:
where a1 is the first term, an is the nth term, and d is the difference between successive terms (e.g. d = 2 in the odd numbers example).
The question says to use the index of the sequence as the domain and the value of the sequence as the range. Here the domain is the list of n's (n = 1, 2, 3, ...) and the range is the an's (a1, a2, a3, ...). This is a function, and it corresponds to a line. See for yourself by graphing the odd numbers example: graph the points (1, 1), (2, 3), (3, 5), (4, 7). What is the slope of the line?
A geometric sequence is a sequence created by multiplying each preceding term by a constant. Take for example: 1, 3, 9, 27, ... which are formed by multiplying each term by 3, starting with 1.
The general equation for a geometric sequence is:
where a is a scale factor, an is the nth term, and r is the ratio between successive terms (our example has a = 1 and r = 3).
Again, the domain is the n's (n = 1, 2, 3, ...) and the range is the an's (a1, a2, a3, ...). This is similar to an exponential function. See this by substituting x for n and y for an in the geometric sequence: y = arx-1. Graph that function and see what it looks like.
Here are two examples of "real life" things that correspond to series/sequences. Which is arithmetic and which is geometric?
A bacterium divides every hour, so that at time 0, there is 1 bacterium, at 1 hour there are 2 bacteria, at 2 hours there are 4, at 3 hours there are 8, etc.
You look at the page numbers of only the right-hand pages of a book. These are 2, 4, ...
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