The problem gives the total cost function for a product as $C(x) = 850\ln(x+10) + 1700$, 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 a total cost of $8500 and round the answer to the nearest whole number.

1. Problem Description

The problem gives the total cost function for a product as $C(x) = 850\ln(x+10) + 1700$, 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 a total cost 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, we find that $\ln(310) \approx 5.73669$.

$C(300) = 850(5.73669) + 1700$

$C(300) = 4876.1865 + 1700$

$C(300) = 6576.1865$

Rounding to the nearest cent, we get $6576.19$.

(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, we find that $e^8 \approx 2980.957987$.

$x = 2980.957987 - 10$

$x = 2970.957987$

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

3. Final Answer

(a) $6576.19

(b) $2971$