What is the smallest perimeter possible for a rectangle whose area is 16 in2, and what are its dimensions?

A = 4,

we denote one side of the rectangle with
a
, and the other with
b
we can write, that:
a
⋅
b
=
16

so we can write, that
b
=
16
a
Now we can write perimeter
P
as a function of
a
P
=
2
⋅
(
a
+
16
a
)
We are looking for the smallest perimeter, so we have to calculate derivative:
P
(
a
)
=
2
a
+
32
a
P
'
(
a
)
=
2
+
(
−
32
a
2
)
P
'
(
a
)
=
2
−
32
a
2
=
2
a
2
−
32
a
2
The extreme values can only be found in points where
P
'
(
a
)
=
0
P
'
(
a
)
=
0
⇔
2
a
2
−
32
=
0
2
a
2
−
32
=
0

x
a
2
−
16
=
0

×
x
.
.
a
2
=
16

×
×
x
a
=
−
4
or
a
=
4
Since, length is a scalar quantity, therefore, it cannot be negative,
When
a
=
4
,
b
=
16
4

b
=
4

The second and fourth answer are correct