The total cost function for a product is given by $C(x) = 850 \ln(x+10) + 1700$, where $x$ is the number of units produced. (a) We need to find the total cost of producing 300 units and round the answer to the nearest cent. (b) We need to find how many units will give total costs of $8500 and round the answer to the nearest whole number.

1. Problem Description

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

(a) We need to find the total cost of producing 300 units and round the answer to the nearest cent.

(b) We need to find how many units will give total costs of $8500 and round the answer 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$:

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

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, $e^8 \approx 2980.957987$

$x = 2980.957987 - 10$

$x = 2970.957987$

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

3. Final Answer

(a) $6576.08$

(b) $2971$