(1 point) For the given cost function $$ C(x)=72900+500 x+x^{2} $$ find: a) The cost at the production level 1300 b) The average cost at the production level 1300 c) The marginal cost at the production level 1300 $$ \square $$ d) The production level that will minimize the average cost e) The minimal average cost $$ \square $$

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**a) Cost at production level 1300**

The cost function is
$$
C(x)=72900+500x+x^2.
$$
For $$ x=1300 $$:
$$
C(1300)=72900+500(1300)+(1300)^2.
$$
Calculate each term:
- $$500(1300)=650000$$,
- $$(1300)^2=1690000$$.

Thus,
$$
C(1300)=72900+650000+1690000=2\,412\,900.
$$

---

**b) Average cost at production level 1300**

The average cost $$ A(x) $$ is given by
$$
A(x)=\frac{C(x)}{x}.
$$
At $$ x=1300 $$:
$$
A(1300)=\frac{2\,412\,900}{1300}\approx 1856.846.
$$

---

**c) Marginal cost at production level 1300**

The marginal cost is the derivative of the cost function:
$$
C'(x)=\frac{d}{dx}\left(72900+500x+x^2\right)=500+2x.
$$
At $$ x=1300 $$:
$$
C'(1300)=500+2(1300)=500+2600=3100.
$$

---

**d) Production level that minimizes the average cost**

The average cost function is
$$
A(x)=\frac{C(x)}{x}=\frac{72900}{x}+500+x.
$$

To minimize $$ A(x) $$, differentiate with respect to $$ x $$:
$$
A'(x)=-\frac{72900}{x^2}+1.
$$

Set the derivative equal to zero:
$$
-\frac{72900}{x^2}+1=0.
$$
This implies
$$
\frac{72900}{x^2}=1 \quad \Longrightarrow \quad x^2=72900.
$$
Taking the positive square root (since production level must be positive):
$$
x=\sqrt{72900}=270.
$$

---

**e) Minimal average cost**

Substitute $$ x=270 $$ into the average cost function:
$$
A(270)=\frac{72900}{270}+500+270.
$$
Calculate:
- $$\frac{72900}{270}=270$$,

Thus,
$$
A(270)=270+500+270=1040.
$$
**a) Cost at production level 1300:** \$2,412,900 **b) Average cost at production level 1300:** Approximately \$1,856.85 **c) Marginal cost at production level 1300:** \$3,100 **d) Production level that minimizes the average cost:** 270 units **e) Minimal average cost:** \$1,040

(1 point) For the given cost function find:
a) The cost at the production level 1300
b) The average cost at the production level 1300
c) The marginal cost at the production level 1300
d) The production level that will minimize the average cost
e) The minimal average cost

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(1 point) For the given cost function find: a) The cost at the production level 1300 b) The average cost at the production level 1300 c) The marginal cost at the production level 1300 d) The production level that will minimize the average cost e) The minimal average cost