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mamaluj [8]
3 years ago
15

Monthly sales of a particular personal computer are expected to decline at the following rate of S'(t) computers per​ month, whe

re t is time in months and​ S(t) is the number of computers sold each month.
S'(t)= -30t^(2/3)

The company plans to stop manufacturing this computer when monthly sales reach 1,000 computers. If monthly sales now ​(t=​0) are 2,440 computers, find​ S(t). How long will the company continue to manufacture this​ computer?
Mathematics
1 answer:
Len [333]3 years ago
6 0

Answer:

S(t) = -18t^\frac{5}{3}+2440

Company will take approximately 14 months

Step-by-step explanation:

Given function that shows the change rate of computers,

S'(t)=-30 t^{\frac{2}{3}}

Where,

t = number of months

On integrating,

\int S'(t) = -30\int t^{\frac{2}{3}} dt

S(t) = -30(\frac{t^{\frac{2}{3}+1}}{\frac{2}{3}+1})+C

S(t) = -30 (\frac{t^{\frac{2+3}{3}}}{\frac{2+3}{3}})+C

S(t) = -30(\frac{t^\frac{5}{3}}{\frac{5}{3}})+C

S(t) = -30(\frac{3t^\frac{5}{3}}{5})+C

S(t) =-18t^\frac{5}{3}+C

According to the question,

S(0) = 2,440,

\implies -18(0)^\frac{5}{3}+C=2440\implies C = 2440

Hence, the required function,

S(t) = -18t^\frac{5}{3}+2440

If S(t) = 1,000,

-18t^\frac{5}{3}+2440=1000

-18t^\frac{5}{3}=-1440

t^\frac{5}{3}=80

t = (80)^\frac{3}{5}\approx 13.86

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