Respuesta :
The length and the width of the rectangle are 6 inches and 6 inches respectively, and the maximum area of a rectangle is 36 square inches if its perimeter is 24 inches
It is given that the perimeter of a rectangle is 24.
It is required to find the dimensions and maximum area of a rectangle.
It is a question about finding maxima.
What are maxima and minima?
Maxima and minima of a function are the extrema within the range, in other words, the maximum value of a function at a certain point is called maxima and the minimum value of a function at a certain point is called minima.
We know the perimeter of a rectangle (p) = 2(l+w)
and area of a rectangle (a) = l×w
Where l = length and w = width
We have p = 2(l+w) = 24 ⇒ l+w = 12 ⇒ w = 12 - l
and a = l×w
or, a = l(12--l)
or, [tex]\rm a = 12 l - l^2[/tex]
To find the maxima and minima of any function we differentiate the function and equate it to zero.
a' = 12- 2l
a' = 0 ⇒ 12 - 2l = 0 ⇒ l = 6 inch
Since a'' ( double derivative) is less than zero ie. [tex]\rm a^{''} \leq 0[/tex] hence the maximum area of the rectangle will be at l = 6 inch.
⇒ w = 12 - l ⇒ 12 - 6 = 6 inch
and the maximum area will be,
a = l × w = 6 ×6 = 36 inch sqaure.
Thus, the length and the width of the rectangle are 6 inches and 6 inches respectively, and the maximum area of a rectangle is 36 square inches if its perimeter is 24 inches
Know more about the maxima and minima here:
brainly.com/question/6422517
