GP Unit 4
Name:
Section Number:
Instructions:
Identify the document by typing your full name and section number next to the yellow text.
Rename the file by adding your last names to current file name (e.g., “u4gp_lastnames.doc”).
Type your answers next to the yellow text.
Unless otherwise stated, answers requiring decimal places should be rounded to two decimals. This includes rounding dollar and cent amounts to the nearest cent.
To show your work, you will need to include
o the algebra used to compute the solution to any equations
o the formula with substituted values.
o the final calculated answer with units.
· To utilize the scientific calculator on your computer, do the following:
o Open the calculator (if it is not in the accessories folder, then select Run from the Start menu)
o Select View from the drop down menu
o Select Scientific to utilize the calculator.
o Note that x^y computes any number to any power (integer, fraction, decimal).
Please add your file.
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.
a) Find the function V that represents the volume of the box in terms of x.
Answer V = 4×3 – 28×2 + 48x
V = (L) (W) (x)
but L = 8 – 2x
W = 6 – 2x
Then V = (8-2x) (6-2x) x
= 48x – 12×2 – 16×2 + 4×3
= 4×3 – 28×2 + 48x
b) Graph this function and show the graph over the valid range of the variable x..
Show Graph here
c) Using the graph, what is the value of x that will produce the maximum volume?
Answer x = 1
Based on the graph the maximum value of x that will produce the maximum volume is 1. This can also be proven by substituting different values to x in the equation V = 4×3 – 28×2 + 48x.
when x = 0.5
V = 4 (0.5)3 – 28(0.5)2 + 48(0.5)
V = 17.5
when x = 1
V = 24
when x = 2
V= 16
when x = 3
V = 0
2) The volume of a cylinder (think about the volume of a can) is given by V = πr2h 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.
a) Write h as a function of r. Keep “p” in the function’s equation.
Answer h = 100/ πr2
V = πr2h
100= πr2h
therefore h = 100/ πr2
b) What is the measurement of the height if the radius of the cylinder is 2 centimeters? Round your answer to the nearest whole number.
Answer 8 cm.
Show work in this space
h = 100/ πr2
h = 100 / π (2)2
h = 8 cm
c) Graph this function.
Show graph here
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 ,000 for 2 years at a rate of 10%.
d) Calculate the return (A) if the bank compounds annually (n = 1). Round your answer to the hundredth’s place.
a)
Answer: $ 12,100
Show work in this space. Use ^ to indicate the power.
A= 10,000[1+ (0.1/1)] ^ (1)(2)
= $ 12,100
b) Calculate the return (A) if the bank compounds quarterly (n = 4). Round your answer to the hundredth’s place.
Answer: $ 12,184.03
Show work in this space
A= 10,000[1+ (0.1/4)] ^ (4) (2)
= $ 12,184.03
c) Calculate the return (A) if the bank compounds monthly (n = 12). Round your answer to the hundredth’s place.
Answer: $ 12, 2043.91
Show work in this space
A= 10,000[1+ (0.1/12)] ^ (12) (2)
= $ 12, 2043.91
d) Calculate the return (A) if the bank compounds daily (n = 365). Round your answer to the hundredth’s place.
Answer: $ 12,213.69
Show work in this space
A= 10,000[1+ (0.1/365)] ^ (365) (2)
= $ 12,213.69
f) What observation can you make about the size of the increase in your return as your compounding increases more frequently?
Answer:As the compounding increases more frequently, the amount of return also increases.
If a bank compounds continuously, then the formula takes a simpler, that is
where e is a constant and equals approximately 2.7183.
Calculate A with continuous compounding. Round your answer to the hundredth’s place.
Answer: ,214.44
Show work in this space
= 10,000 (2.7183) ^ (0.1) (2)
= ,214.44
g) Now suppose, instead of knowing t, we know that the bank returned to us ,000 with the bank compounding continuously. Using natural logarithms, find how long we left the money in the bank (find t). Round your answer to the hundredth’s place.
Answer: 2.71 years
Show work in this space
g)
15,000 = 10,000e0.1t
15,000/10,000 = e0.1t
1.25 = e0.1t
ln 1.5 = ln e0.1t
ln 1.5 = 0.1t
t = ln 1.5/ 0.1
t = 2.71 years
h) A commonly asked question is, “How long will it take to double my money?” At 10% interest rate and continuous compounding, what is the answer? Round your answer to the hundredth’s place.
Answer: 6.93 years ≈ 7 years
Show work in this space
t @ A = 2P
using
2P = Pe0.1t
2 = e0.1t
ln 2 = 0.1t
t = (ln 2)/ 0.1
t = 6.93 years ≈ 7 years
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.
a) Show coordinates in this space
t = 0, 1, 2, 3, 4
A =1, 1.1, 1.21, 1.33, 1.46
Show work in this space
using ; n = 1; r = 8%; P = 20,000
@ t = 0
A = 1 [1.1] ^0
A= 1
@t = 1
A = 1[1.1] ^ 1
A = 1.1
@ t = 2
A = 1 [1.1] ^ 2
A = 1.21
@ t = 3
A = 1 [1.1] ^ 3
A = 1.33
@ t = 4
A = 1 [1.1] ^ 4
A = 1.46
b) Show graph here
5) Logarithms:
a) Using a calculator, find log 10000 where log means log to the base of 10.
Answer: 4
b) Most calculators have 2 different logs on them: log, which is based 10, and ln, which is based e. In computer science, digital computers are based on the binary numbering system which means that there are only 2 numbers available to the computer, 0 and 1. When a computer scientist needs a logarithm, he/she uses a log to the base 2. To find the log of a number to any base, we can use a conversion formula as shown here:
Using this formula, find . Round your answer to the hundredth’s place.
Answer: 13.288
Show work in this space
log2 10000 = log 10000/ log 2
= 13.288
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. Include the following items in your posting:
· The formula for measuring sound.
· Pick a specific sound, give the decibels of the sound, and explain what this measurement means.
Answer:
The loudness of a sound is dependent on the power of sound in watts and on the area of the listener’s location. That is sound can be measured using the formula: Power (watts)/ area (m2 or cm2). To determine the intensity of the sound or how loud a sound is, sounds are measured relative to a standard threshold of hearing which can be indicated by Io and is equal to 10-12 watts/ m2 or 10-16 watts/ cm2. This standard threshold is the weakest sound the ear can hear. Lower than this value can not be heard by human ear.
It is also common to express sound in decibel (dB). To measure the intensity of sound in decibel, the formula below is used:
I (dB) = 10 log 10[ I/ Io]
With this formula, we can say that the standard threshold of hearing is 0dB because:
I (dB) = 10 log 10[(10-12 watts/ m2) / (10-12 watts/ m2)]
= 10 log 10 (1)
= 0dB
A rock concert is measured to have 120 dB, which is also the threshold of pain or the sound that is unbearable to the listener for a long period. Some say the threshold of pain is at 130dB and a 150dB sound may cause breakage of the bones in the ear (Schoolscience, 2006). To be safe, we will say that the threshold of pain is between, 115- 135 dB. The intensity of a sound increases by a factor of 10 for every 10dB increase (Anonymous, 2006). For example, a whisper has 30dB intensity while a normal conversation has 50dB intensity. This means that a normal conversation which is 20db more than a whisper is 100 times as intense as the whisper. So the loudness of a rock concert which is 120dB means that 1012 times as intense as the threshold of hearing and 10,000,000 times as intense as a normal conversation and a whisper may never be heard at a rock concert.
Credit:ivythesis.typepad.com
0 comments:
Post a Comment