a company introduces a new product for which the number of units sold S is
S(t)=200(5-(9/2+t))
where t is the time in months.
a.)find the average value of S(t) during the first year.
b.) during what month does S'(t) equal the average value during the first year?
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To find the average value of a function over a specific interval, we need to calculate the definite integral of the function over that interval and divide it by the length of the interval.
a.) To find the average value of S(t) during the first year (12 months), we need to calculate the definite integral of S(t) from t = 0 to t = 12 and divide it by 12.
Let's find the definite integral of S(t) first:
∫[0 to 12] S(t) dt = ∫[0 to 12] 200(5-(9/2+t)) dt
Simplifying the integrand:
= 200 ∫[0 to 12] (5-(9/2+t)) dt
= 200 ∫[0 to 12] (5 - 9/2 - t) dt
= 200 ∫[0 to 12] (1/2 - t) dt
= 200 [1/2t - (t^2)/2] evaluated from 0 to 12
= 200 [(1/2 * 12 - (12^2)/2) - (1/2 * 0 - (0^2)/2)]
= 200 [(6 - 72/2) - 0]
= 200 [6 - 36]
= 1200 - 720
= 480
Now, divide the result by 12 to find the average value:
Average value of S(t) = 480 / 12 = 40
Therefore, the average value of S(t) during the first year is 40.
b.) To find the month when S'(t) (the derivative of S(t)) equals the average value during the first year, we need to solve the equation S'(t) = 40.
First, let's calculate S'(t) by taking the derivative of S(t):
S'(t) = dS(t)/dt
= d/dt (200(5-(9/2+t)))
= -200(1)
Setting S'(t) equal to 40:
-200(1) = 40
Simplifying the equation:
-200 = 40
This equation has no solution. Therefore, there is no month when S'(t) equals the average value during the first year.
Please note that it's important to check if there are any errors or typos in the given equation or if any additional constraints or information need to be considered for a more accurate solution.