The problem gives a cost function $C(x) = 850 \ln(x + 10) + 1700$, where $x$ is the number of units produced. Part (a) asks to find the total cost of producing 300 units, rounded to the nearest cent. Part (b) asks how many units will give total costs of $8500, rounded to the nearest whole number.

1. Problem Description

The problem gives a cost function $C(x) = 850 \ln(x + 10) + 1700$, where $x$ is the number of units produced. Part (a) asks to find the total cost of producing 300 units, rounded to the nearest cent. Part (b) asks how many units will give total costs of $8500, rounded to the nearest whole number.

2. Solution Steps

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

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

Using a calculator, $\ln(310) \approx 5.73657$.

Then $C(300) \approx 850(5.73657) + 1700 \approx 4876.0845 + 1700 \approx 6576.0845$.

Rounding to the nearest cent, we get $6576.08$.

(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 with base $e$:

$e^8 = x + 10$.

$x = e^8 - 10$.

Using a calculator, $e^8 \approx 2980.957987$.

Then $x \approx 2980.957987 - 10 \approx 2970.957987$.

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

3. Final Answer

(a) $6576.08

(b) 2971