a. Find the dimensions of the rectangle with the greatest area that can be built
ID: 2850095 • 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 less than equal to x less than equal to 1) and one corner of the rectangle is on the curve, y = x^3 . What is the area of this rectangle? (Hint: The area function is not simply A(x) = x^4) b. Generalize the problem in part (a) for the curve y = Cx^3 with C > 0 and 0 less than equal to x less than equal to 1. c. Generalize for the curve y = Cx^3 with C > 0 and 0 less than equal to x less than equal to B. d. Generalize for the curve y = Cx^n with C > 0, n a positive integer, and 0 less than equal to x less than equal to B.Explanation / Answer
a.
Suppose length of rectangle be a,
Height of rectangle (1-a)^3
Area, A= a*(1-a)^3
dA/da = -3a(1-a)^2+(1-a)^3 = 0
a = 1, 0.25
Thus, Area (max) = 0.25*0.75^3 = 0.105
b.
In this case
Height of rectangle C(1-a)^3
Area, A= Ca*(1-a)^3
dA/da = C[-3a(1-a)^2+(1-a)^3] = 0
a = 1, 0.25
Thus, Area (max) = 0.25*0.75^3*C = 0.105C
c.
In this case
Height of rectangle C(B-a)^3
Area, A= Ca*(B-a)^3
dA/da = C[-3a(B-a)^2+(B-a)^3] = 0
a = B, 0.25B
Thus, Area (max) = 0.25*0.75^3*B^3*C = 0.105(B^3)C
d.
In this case
Height of rectangle C(B-a)^n
Area, A= Ca*(B-a)^n
dA/da = C[-3a(B-a)^(n-1)+(B-a)^n] = 0
a = B, 0.25B
Thus, Area (max) = 0.25*0.75^3*B^n*C = 0.105(B^n)C
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