Answer:
the solution to the problem is
(
a
,
b
)
=
(
24
,
35.75
)
Step-by-step explanation:
Let a be the width of the poster and b the height.
Let A be the area of the poster to be minimized.
A
=
380+
2
(
a
⋅
4
)
+
2
(
b
⋅
6
)
−
4
(
6
⋅
4
)
=
3840+
8
a+
12
b
−
96
=
284+
8
a
+
12
b
A
=
a
⋅
b
So let's get a in function of b:
284
+
8
a
+
12
b
=
a
b
284
+
12
b
=
a
b
−
8
a
284
+
12
b
=
a
(
b
−
8
)
a
=284+
12
b/
b
−
8
Now,
A
(
b
)
will be the function in one single variable (b) that we will minimize:
A
(
b
)
=
a
⋅
b
=
(
284
+
12
b
b
−
8
)
⋅
b
=
284
b
+
12
b
^3/
b
−
8
I have to find the first derivative of the function to minimize it:
A
'
(
b
)
=
12
b^
2
−
16
b
−
192/
(
b
−
8
)^
2
The minimum points satisfy the condition
A
'
(
b
)
=
0
so:
b
2
−
16
b
−
192
=
0
b
=
−
8 or b
=
24
But for obvious reason (b is the height of a poster), b must be positive, so only b=24 is a correct solution.
Now we have to find a:
a
=
284+
12
b/
b
−
8
=
284+
12
⋅
24
/24
−
8
=
35.75
So, the solution to the problem is
(
a
,
b
)
=
(
24
,
35.75
)