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.
- Type your answers next to the yellow text.
- To show your work, you will need to include
o the formula with substituted values.
o the final calculated answer with units.
Please submit your assignment.
1) An open-top box is to be constructed from a 4 by 6 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 – 20×2 + 24x
V = (L) (W) (x)
but L = 6 – 2x
W = 4 – 2x
Then V = (6-2x) (4-2x) x
= 24x – 12×2 – 8×2 + 4×3
= 4×3 – 20×2 + 24x
a) Graph this function.
FORMTEXT Show Graph here.
b) Using the graph, what is the value of x that will produce the maximum volume?
Answer. x = 1
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 121 cubic centimeters.
a) Write h as a function of r.
Answer: h = 121/ πr2
Show work in this space.
V = πr2h
121= πr2h
therefore h = 121/ πr2
b) What is the measurement of the height if the radius of the cylinder is 3 centimeters?
Answer: 4.28 cm
Show work in this space.
h = 121/ πr2
= 121/ π (32)
h = 4.28 cm
c) Graph this function.
Show graph here.
3) The formula for calculating the amount of money returned for deposit money into a bank account or CD (Certificate of Deposit) is given by the following:
A is the amount of returned
P is the principal amount 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 3 years at a rate of 8%.
a) Calculate the return (A) if the bank compounds annually (n = 1).
Answer: $ 25,194.24
Show work in this space. Use ^ to indicate the power.
A= 20,000[1+ (0.08/1)] ^ (1)(3)
= $ 25,194.24
b) Calculate the return (A) if the bank compounds quarterly (n = 4).
Answer: , 364.83
Show work in this space .
A = 20,000[1+ (0.08/4)]^ (4)(3)
= ,364.83
c) Calculate the return (A) if the bank compounds monthly (n = 12).
Answer: ,404.74
Show work in this space.
A = 20,000 [1+ (0.08/12)]^ (12)(3)
= ,404.74
d) Calculate the return (A) if the bank compounds daily (n = 365).
Answer: ,424.31
Show work in this space.
A = 20,000 [1+ (0.08/365)]^ (365)(3)
= ,424.31
e) What observation can you make about the increase in your return as your compounding increases more frequently?
Answer: As the compounding increases more frequently, the amount of return also increases.
f) If a bank compounds continuous, then the formula becomes simpler, that is
where e is a constant and equals approximately 2.7183. Calculate A with continuous compounding.
Answer: ,425.02
Show work in this space
= 20,000 (2.7183)^ (0.08)(3)
= 25,425.02
g) Now suppose, instead of knowing t, we know that the bank returned to us ,000 with the bank compounding continuously. Using logarithms, find how long we left the money in the bank (find t).
Answer: 2.79 years
Show work in this space
25,000 = 20,000e0.08t
25,000/20,000 = e0.08t
1.25 = e0.08t
ln 1.25 = ln e0.08t
ln 1.25 = 0.08t
t = ln 1.25/ 0.08
t = 2.79 years
h) A commonly asked question is, “How long will it take to double my money?” At 8% interest rate and continuous compounding, what is the answer?
Answer: 8.66 years ≈ 9 years
Show work in this space.
t @ A = 2P
using
2P = Pe0.08t
2 = e0.08t
ln 2 = 0.08t
t = (ln 2)/ 0.08
t = 8.66 years ≈ 9 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 = 8%, P = 1, and n = 1 and give the coordinates (t,A) for the points where t = 0, 1, 2, 3, 4.
a) Show coordinates in this space.
t = 0, 1, 2, 3, 4
A =,000; ,600; ,328; ,194.24; ,209. 78
Show work in this space.
using ; n = 1; r = 8%; P = 20,000
@ t = 0
A = 20,000 [1.08] ^0
A= 20,000
@t = 1
A = 20,000[1.08] ^ 1
A = 21,600
@ t = 2
A = 20,000 [1.08] ^ 2
A = 23,328
@ t = 3
A = 20,000 [1.08] ^ 3
A = 25,194.24
@ t = 4
A = 27,209.78
b) Show graph here.
1) Logarithms:
a) Using a calculator, find log 1000 where log means log to the base of 10.
Answer: 3
b) Most calculators have 2 different logs on them: log, which is base 10, and ln, which is base 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 needs a log to base 2 which is not on any calculator. To find the log of a number to any base, we can use a conversion formula as shown here:
Using this formula, find .
Answer: 9.965784285
Show work in this space.
log2 1000 = log 1000
log 2
= 3/ 0.301
= 9.965784285
Credit:ivythesis.typepad.com
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