Question

The total cost function for a product is C(x) = 750 ln(x + 10) + 1900

Answer 1

Answer:

(a) $6,412.12

(b) 6624.

Step-by-step explanation:

We have been given the total cost function for a product is $C(x)=750\text{ ln}(x+10)+1900$, where x is the number of units produced.

(a) To find the total cost of producing 400 units, we will substitute $x=400$ in our given formula.

$C(400)=750\text{ ln}(400+10)+1900$  

$C(400)=750\text{ ln}(410)+1900$  

$C(400)=750*6.0161571596983535+1900$  

$C(400)=4512.117869773765125+1900$  

$C(400)=6412.117869773765125$  

$C(400)\approx 6412.12$  

Therefore, the total cost of producing 400 units is $6,412.12.

(b) To find the number of units produced with total costs of $8500, we will substitute $C(x)=8500$ in our given formula.

$8500=750\text{ ln}(x+10)+1900$  

Switching sides:

$750\text{ ln}(x+10)+1900=8500$

Subtract 1900 from both sides:

$750\text{ ln}(x+10)+1900-1900=8500-1900$  

$750\text{ ln}(x+10)=6600$

Now, we will divide both sides of our equation by 750.

$\frac{750\text{ ln}(x+10)}{750}=\frac{6600}{750}$  

$\text{ ln}(x+10)=\frac{44}{5}$  

Using logarithm definition, we will get:

$x+10=e^{\frac{44}{5}}$  

$x+10=6634.244006277887$  

$x+10-10=6634.244006277887-10$  

$x=6624.244006277887$  

$x\approx 6624$  

Therefore, producing 6624 units will give total costs of $8500.