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

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. We need to find:

(a) the total cost of producing 300 units, rounded to the nearest cent.

(b) the number of units that will give a total cost 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:

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

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

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

$C(300) = 850(5.73657) + 1700$

$C(300) = 4876.0845 + 1700$

$C(300) = 6576.0845$

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$:

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

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

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

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

$8 = \ln(x + 10)$

To solve for $x$, we take the exponential of both sides:

$e^8 = x + 10$

$x = e^8 - 10$

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

$x = 2980.957987 - 10$

$x = 2970.957987$

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

3. Final Answer

(a) $6576.08$

(b) $2971$