Calculus-Applied Max/Min

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, y=x^3
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.

Answer 1

Say, the dimensions of the rectangle are

Length along x axis=s

Width perpendicular to x axis=t

 

Say, the rectangle starts at a distance x from origin on x axis.

So.  t=x^3                   using y=x^3

with above points in mind

s=(1-x)

The area of the rectangle is

A =s*t

   =(1-x)*x^3

dA/dx= 3x^2-4x^3=0

Either x= 0   or.  x=3/4

So, s=1-x=1-(3/4)=1/4

t=x^3=(3/4)^3=27/64

So, dimensions of the rectangle are 1/4. and. 27/64

Area of the rectangle= s*t=27/256

 

Parts b , c , and d can likewise be extended further.

Answer 2

As written, this problem set doesn't say which top side corner has to be on the graph of the given equations.  The answers which yield the largest rectangle areas will be those for which the corner with the higher x value will lie on the function graph.  In this case, the horizontal side length will be 1 for cases "a" and "b", and "B" for the other two cases.  The vertical side lengths will be 1 for case "a", "C" for case "b", "CB³" for case "c", and "CBⁿ" for case "d".  Thus, the maximum areas will be 1, C, CB⁴, and CBⁿ⁺¹, respectively.

If the top side corner that must be on the function graph is the one with the smaller x value, then the problem becomes more interesting...