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?
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.