A rectangle has dimensions 3x and 5-2x. 1. What is the maximum area of the rectangle? 2. What value of 'x' gives the maximum area?

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jeew-m | College Teacher | (Level 1) Educator Emeritus

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Area of rectangle;

`A = 3x(5-2x) = 15x-6x^2`

 

For maximum or minimum area `(dA)/dx = 0`

`(dA)/dx = 15-12x`

When `(dA)/dx = 0`

`15-12x = 0`

        `x = 5/4`

 

If A has a maximum then `(d^2A)/dx^2 < 0` at x = 5/4

 

`(dA^2)/dx^2 = -12 < 0 `

 

So A has a maximum.

Maximum `A = 3*5/4(5-2*5/4) = 9.375`

 

So the maximum area of the rectangle is 9.375 and it occurs when x = 5/4.

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