Respuesta :
Answer:
(a) $6,412.12
(b) 6624.
Step-by-step explanation:
We have been given the total cost function for a product is [tex]C(x)=750\text{ ln}(x+10)+1900[/tex], where x is the number of units produced.
(a) To find the total cost of producing 400 units, we will substitute [tex]x=400[/tex] in our given formula.
[tex]C(400)=750\text{ ln}(400+10)+1900[/tex]
[tex]C(400)=750\text{ ln}(410)+1900[/tex]
[tex]C(400)=750*6.0161571596983535+1900[/tex]
[tex]C(400)=4512.117869773765125+1900[/tex]
[tex]C(400)=6412.117869773765125[/tex]
[tex]C(400)\approx 6412.12[/tex]
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 [tex]C(x)=8500[/tex] in our given formula.
[tex]8500=750\text{ ln}(x+10)+1900[/tex]
Switching sides:
[tex]750\text{ ln}(x+10)+1900=8500[/tex]
Subtract 1900 from both sides:
[tex]750\text{ ln}(x+10)+1900-1900=8500-1900[/tex]
[tex]750\text{ ln}(x+10)=6600[/tex]
Now, we will divide both sides of our equation by 750.
[tex]\frac{750\text{ ln}(x+10)}{750}=\frac{6600}{750}[/tex]
[tex]\text{ ln}(x+10)=\frac{44}{5}[/tex]
Using logarithm definition, we will get:
[tex]x+10=e^{\frac{44}{5}}[/tex]
[tex]x+10=6634.244006277887[/tex]
[tex]x+10-10=6634.244006277887-10[/tex]
[tex]x=6624.244006277887[/tex]
[tex]x\approx 6624[/tex]
Therefore, producing 6624 units will give total costs of $8500.
