a. Find the dimensions of the rectangle with the greatest area that can be built
ID: 2833682 • Letter: A
Question
a. Find the dimensions of the rectangle with the greatest area that can be built so the base of the rectangle is on the x-axis between 0 and 1 (0 <= x <= 1) and one corner of the rectangle is on the curve = x^3. What is the area of this rectangle? b. Generalize the problem in part (a) for the curve y =Cx^3 with C > 0 and 0 <= x <= 1. c. Generalize for the curve y =Cx^3 with C > 0 and 0 <= x <= B. d. Generalize for the curve y =Cx^n with C > 0, n a positive integer, and 0 <= x <= B.
Explanation / Answer
(a)
area A= x*x^3 = x^4
=>
A' = 4x^3 >=0=> A is increasing
=>
for maximum area x = 1
=> dimensions are (1,1)
(b)
A = x*cx^3 = cx^4
=>
A' = 4Cx^3>0
=> A is increasing
=> for maximum area x = 1
=>
dimensions are (1,C)
(c)
following above logic
dimensions are (B,CB^3)
(d)
dimensions are (B,CB^n)
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