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# Linear and Non-Linear Equations : Finding Minimum or Maximum Value

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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.

b) Graph this function.
Show Graph here.

c) Using the graph, what is the value of x that will produce the maximum volume?

2) The volume of a cylinder (think about the volume of a can) is given by V = &#960;r2h 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.
Show work in this space.

b) What is the measurement of the height if the radius of the cylinder is 3 centimeters?
Show work in this space.

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 \$20,000 for 3 years at a rate of 8%.
a) Calculate the return (A) if the bank compounds annually (n = 1).
Show work in this space. Use ^ to indicate the power.

b) Calculate the return (A) if the bank compounds quarterly (n = 4).
Show work in this space .

c) Calculate the return (A) if the bank compounds monthly (n = 12).
Show work in this space.

d) Calculate the return (A) if the bank compounds daily (n = 365).
Show work in this space.

e) What observation can you make about the increase in your return as your compounding increases more frequently?

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.
Show work in this space

g) Now suppose, instead of knowing t, we know that the bank returned to us \$25,000 with the bank compounding continuously. Using logarithms, find how long we left the money in the bank (find t).
Show work in this space

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: Show work in this space.

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.

Show work in this space.

b) Show graph here.

5) Logarithms:
a) Using a calculator, find log 1000 where log means log to the base of 10.

https://brainmass.com/math/linear-algebra/linear-and-non-linear-equations-finding-minimum-or-maximum-value-70424

#### Solution Preview

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.

The area of the base is: (4-2x)*(6-2x)
The height is: x, so the volume V = (4-2x)*(6-2x) *x

b). Graph this function.
You can draw it according to the function. Pay attention that two points (2, 0) (0, 0). i.e, when
x = 2, 0, the volume is 0 and also x , V > 0, i.e. x cannot be negative, and the volume cannot be negative. The range of x is 0<x< 2, because (4-2x) > 0

c). From the graph, we can see that at the point where V is maximum, then the tangent of point(x,V) is 0, i.e, dV/dx = 0 at point (x, V)

dV/dx = d((4-2x)*(6-2x) *x))/dx = ...

#### Solution Summary

Equations are written from word problems and solved for optimum value.

\$2.19