Final answer:
To find the average cost function and the average cost of 16 units, we integrate the rate of change of the average cost function and substitute x = 16 into the average cost function, respectively.
Step-by-step explanation:
To find the average cost function
�
ˉ
(
�
)
C
ˉ
(x), you need to integrate the given rate of change function C^{\bar{}}'(x) with respect to
�
x. The given rate of change is C^{\bar{}}'(x) = -4x^2 - 2x + 41.
Let's find
�
ˉ
(
�
)
C
ˉ
(x):
�
ˉ
(
�
)
=
∫
(
−
4
�
2
−
2
�
+
41
)
�
�
C
ˉ
(x)=∫(−4x
2
−2x+41)dx
�
ˉ
(
�
)
=
−
4
3
�
3
−
�
2
+
41
�
+
�
C
ˉ
(x)=−
3
4
x
3
−x
2
+41x+C
Now, you need to find the constant of integration
�
C. The problem states that the average cost of 4 units is $13.00. So, you can use this information to find
�
C:
�
ˉ
(
4
)
=
−
4
3
(
4
)
3
−
(
4
)
2
+
41
(
4
)
+
�
=
13
C
ˉ
(4)=−
3
4
(4)
3
−(4)
2
+41(4)+C=13
Now, solve for
�
C:
−
4
3
(
64
)
−
16
+
164
+
�
=
13
−
3
4
(64)−16+164+C=13
−
256
3
+
148
+
�
=
13
−
3
256
+148+C=13
�
=
13
+
256
3
−
148
C=13+
3
256
−148
�
=
5
3
C=
3
5
Now that you have
�
C, the average cost function
�
ˉ
(
�
)
C
ˉ
(x) is:
�
ˉ
(
�
)
=
−
4
3
�
3
−
�
2
+
41
�
+
5
3
C
ˉ
(x)=−
3
4
x
3
−x
2
+41x+
3
5
Now, for part (b), find the average cost of 16 units:
�
ˉ
(
16
)
=
−
4
3
(
16
)
3
−
(
16
)
2
+
41
(
16
)
+
5
3
C
ˉ
(16)=−
3
4
(16)
3
−(16)
2
+41(16)+
3
5
Calculate this expression to find the average cost for 16 units. Ensure to round your answer to the nearest cent.