Producing how many units will give total costs of $6500?

Question

The total cost function for a product is

C(x) = 650 ln(x + 10) + 1800

Producing how many units will give total costs of $6500? (Round your answer to the nearest whole number.)

Answer 1

So we need to figure out when total costs equal $6,500. Our first step is to set C(x) equal to $6,500, since C(x) represents our total cost.

6500 = 650ln(x+10) + 1800

Now, we can subtract 1800 from both sides to isolate the 650ln(x+10).

4700 = 650ln(x+10)

Next, we can divide both sides by 650 to isolate ln(x+10). Using a calculator, 4700/650 is about equal to 7.231

7.231 = ln(x+10)

Now is the tricky part. ln is essentially a log of base e. What do we know about logs?

If log_ab = c, a^c = b

So let's apply that here...

7.231 = ln(x+10) is the same thing as 7.231 = log_e(x+10)

Thus...

e^7.231 = x+10

x = e^7.231 - 10

Using a calculator, x = 1371.28, which is 1371 units to the nearest whole number.

Answer 2

6500 = 650 ln(x + 10) + 1800 Subtract 1800 from both sides

6500-1800 = 650 ln(x + 10) + 1800-1800 Simplify

4700 = 650 ln(x + 10) Divide both sides by 650

4700/650 = 650 ln(x + 10)/650 Simplify

94/13=ln(x + 10) Raise both sides using e as the base

e^94/13=e^{ln(x + 10)} Simplify

1381=x+10 Subtract 10 from both sides

1381-10=x+10-10 Simplify

1371 units = x