Question

A rectangle has dimensions 0.7x and 5 - 3x. What value of x gives the maximum area and what is the maximum area?

Answer 1

at `x=0.8dot 3`
`1.458dot3 unit^2`

Explanation:

Given dimensions of the rectangle are: `0.7x and (5 - 3x)`

Area of the rectangle`A=lxxb`, insering given values we obtain
Area of the rectangle`A=0.7x xx (5 - 3x)`,
`A=3.5x-2.1x^2`,
To maximize the area the first derivative is to be set equal to `0`
or `3.5-2.1xx2x=0`, solving for `x` we obtain
`3.5-4.2x=0`
or `x=3.5/4.2=5/6`
`=>x=0.8dot 3`
To confirm that it is a maximum we need to evaluate second derivative at the obtained value of `x`
Second derivative `=-4.2`. Since it is a negative quantity, therefore it is a maxima.

Maximum Area `=0.7xx0.8dot3xx(5-3xx0.8dot3)`
`=0.58dot3xx2.5`
`=1.458dot3 unit^2`

Answer 2

Maximum area = `1.4361" units" ^2` to 4 decimal places

`x=0.35/4.2 = 0.083bar3` to 4 decimal places

(`" "x=2809/3000 " "` as an exact value )

Explanation:

Let the area be `a`

From the question the area is:

`" "a=0.7x(5-3x)`

`" "=> a=0.35x-2.1x^2`

Rate of change in `("change in "a)/("change in "x)`

Using calculus: shortcut method

`(delta a)/(delta x) = 0.35-4.2x`

Rate of change is zero at maximum area

`=> (delta a)/(delta x) = 0 =0.35-4.2x`

`=> 4.2x=0.35`

`=> x=0.35/4.2 = 0.083bar3`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~
to find the fractional solution:

`10x=8.3bar3`

so `10x-x=8.427`

`x=8.427/9 = 8427/9000" "` Mmmmmm!
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Thus the exact area is

`0.7xx 8427/9000xx(5-3xx8427/9000)`

The approximate area is

`3.2772-1.8411=1.4361` to 4 decimal places