A rectangle with its base on the x-axis and its vertices on the curve
y =4 - x² can have a maximum area of 32/3√3.
The formula for calculating a rectangle's area is A=length*breadth
The base's length is x.
y represents the curve's vertices (width).
Given that a rectangle's area increases as its base moves along the x-axis, our base will be 2x.
The rectangle's surface area will change to:
A = 2x ( 4 - x²)
When the area is at its largest, da / dx should be maximum.
A = 8x - 2x³
da/dx = 8 - 6x²
8 - 6x² = 0
6x² = 8
x² = 8/6
x = √4/3
=2/√3
We shall substitute 2/√3 into the region in order to obtain the maximum area.
A= 8x - 2x³
= 8*2/√3 - 2*(2/√3)³
=32/3√3
Therefore, the largest area a rectangle with a base on the x-axis and vertices on the curve y = 4 - x² can have is 32/3√3.
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