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

2 Answers
Mar 29, 2016

Answer:

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#

Mar 29, 2016

Answer:

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