Question

Find the length and width (in feet) of a rectangle that has the given area and a minimum perimeter. Area: 25 square feet

1. Length = 5 ft & width = 5 ft of a rectangle that has the given area .

What is called rectangle?
• A Rectangle is a four sided-polygon, having all the internal angles equal to 90 degrees.
• The two sides at each corner or vertex, meet at right angles.
• The opposite sides of the rectangle are equal in length which makes it different from a square.
Let the length and width of rectangle be x feet and y feet respectively.
Give       area = 25 feet²
x y = 25 feet²
y = 25/x feet²   ………………1
Now let perimeter is p
then    p = 2 (x+y) = 2x + 2y
p = 2x + 2 × 25/x
P = 2X + 50/x
diffrentiating w.r.t. x
p’  = 2 – 50/x² ……………..2
putting p’ = 0 to find critical point
2 –  50/x²   = 0
x²  = 25
x  = 5                         [ x ≠ 5 ∴ length can not be negative ]
Now,    p” = 0 + 100/ x³                       from (2)
checking value of p” at critical point
p” at x = 5        = 100/(x)³         > 0
Hence, perimeter is minimum at x = 5 ft
putting       x = 5  in equation 1
y = 1
Hence,          length = 5 ft
width = 5 ft