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The company will continue to manufacture this model for 19 months.
The question is ill-formatted. The understandable format is given below.
An SUV model's monthly sales are anticipated to rise at a rate of
[tex]S'(t)=-24^{1/3}[/tex] SUVs each month, where "t" denotes the number of months and "S(t)" denotes the monthly sales of SUVs. When monthly sales of 300 SUVs are reached, the business intends to discontinue producing this model.
Find S if monthly sales of SUVs are 1,200 at time t=0 (t). How long will the business keep making this model?
Given [tex]S'(t)=-24^{1/3}[/tex] ... ... (1)
Condition (1) at t=0; s(t) =1200
We find S(t)=300 then t=?
a) We find S(t)
from [tex]S'(t)=-24^{1/3}[/tex]
[tex]\implies \frac{dS(t)}{dt}=-24t^{1/3} ~~[\because X'(t)=\frac{dX}{dt} \\\implies dS(t) = -24t^{1/3}dt\\[/tex]
On Integrating both sides
[tex]S(t)=-18t^{4/3}+c ~...~...~(2)[/tex]
Now, at t=0 then S(t)=1200
So, from (2)
1200=0+c
⇒c=1200
[tex]\therefore[/tex]from eq (2)
[tex]S(t)=-18t^{4/3}+1200 ~...~...~(3)[/tex]
b) Considering that the firm intends to cease production of this model once monthly sales exceed 300 SUVs.
So, take S(t)=300
from eq (2) [tex]300=-18t^{4/3}+1200[/tex]
[tex]t^{4/3}=\frac{1200-300}{18}\\=\frac{900}{18}=50\\\implies t=50^{3/4}\\\implies t=18.8030[/tex]
Hence the company will continue 18.8020 months.
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