The problem provides a total cost function $C(x) = 850 \ln(x + 10) + 1700$, where $x$ is the number of units produced. (a) We need to find the total cost of producing 300 units. (b) We need to find the number of units that will give a total cost of $8500.

1. Problem Description

The problem provides a total cost function $C(x) = 850 \ln(x + 10) + 1700$, where $x$ is the number of units produced.

(a) We need to find the total cost of producing 300 units.

(b) We need to find the number of units that will give a total cost of $

8

5

0

0.

2. Solution Steps

(a) To find the total cost of producing 300 units, we substitute $x = 300$ into the cost function $C(x)$.

$C(300) = 850 \ln(300 + 10) + 1700 = 850 \ln(310) + 1700$

Using a calculator, we find that $\ln(310) \approx 5.7365725$.

$C(300) = 850(5.7365725) + 1700 = 4876.086625 + 1700 = 6576.086625$

Rounding to the nearest cent, we get $6576.09$.

(b) To find the number of units that will give a total cost of $8500, we set$C(x) = 8500$and solve for$x$.

$= 850 \ln(x + 10) + 1700$

Subtract 1700 from both sides:

$8500 - 1700 = 850 \ln(x + 10)$

$6800 = 850 \ln(x + 10)$

Divide both sides by 850:

$\frac{6800}{850} = \ln(x + 10)$

$8 = \ln(x + 10)$

Exponentiate both sides using base $e$:

$e^8 = x + 10$

$e^8 \approx 2980.957987$

$2980.957987 = x + 10$

$x = 2980.957987 - 10$

$x = 2970.957987$

Rounding to the nearest whole number, we get $x = 2971$.

3. Final Answer

(a) $6576.09

(b) 2971