Question 166875
1. Find the dimensions of a rectangle "a" with the greatest area whose perimeter is 30 feet.
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P = 2l +2w
30 = 2l + 2w
15 = l + w
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Area = l(w)
A = w(15-w)
A = 15w - w^2
This is a quadratic with a=-1,b=15,c= -A
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Greatest area occurs when w = -b/2a = -15/(2(-1)) = 7.5 feet (this is the width)
Since L+w = 15, length = 7.5 ft
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2. Given x^3-4x^2+2x+1=0
# of sign changes in f(x) = 2
# of sign changes in f(-x) = 1
(a) How many possible positive roots are there?: 0 or 2
(b) How many possible negative roots are there?: 1
(c) What are the possible rational roots? 1 and -1
(d) Using synthetic substitution, which of the possible rational roots is actually a root of the equation? +1
(e) Find the irrational roots of the equation. (Hint: Use the quadratic formula to solve the depressed equation.) 
1)....1....-4....2....1
-------1-----3....-1..|..0
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Quotient: x^2-3x-1
Roots: x = [3 +- sqrt(9 - 4*1*-1)]/2
x = [3 +- sqrt(13)]/2
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Cheers,
Stan H.