Volume, Interest and Logarithms
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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) " in the function's equation. Write h as a function of r. Keep "
Answer
as interest is compounded more frequently the amount of return increases
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
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c) Graph this function.
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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.
Answer:
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Calculate the return (A) if the bank compounds quarterly (n = 4). Round your answer to the hundredth's place.
Answer:
Calculate the return (A) if the bank compounds monthly (n = 12). Round your answer to the hundredth's place.
Answer:
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Calculate the return (A) if the bank compounds daily (n = 365). Round your answer to the hundredth's place.
Answer:
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What observation can you make about the size of the increase in your return as your compounding increases more frequently?
Answer:
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:
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Now suppose, instead of knowing t, we know that the bank returned to us $15,000 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:
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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:
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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
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b) Show graph here
5) Logarithms:
Using a calculator, find log 10000 where log means log to the base of 10.
Answer:
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:
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