4. A company introduces a new product for which the number...

4. A company introduces a new product for which the number of units sold S is $S(t) = 200 left(5 - frac{9}{2 + t} ight)$ Where $t$ is the time in months. a) Find the average rate of change of S during the first year. b) During what month of the first year does S'(t) equal the average rate of change?

Transcript

00:01 In this question, we are told that the number of units sold after t month equals to is given by the function s.

00:08 And we are asked to find the average rate of change of the function fs during the first year.

00:15 And we are going to call it v average.

00:17 And we are asked to determine when s prime of t, when is s prime of t equal to the average rate of change.

00:28 So first, let's answer the first question.

00:34 And one year, recall that one year equals to 12 months.

00:41 We need this because in the function s, t is time in month, right? not in years.

00:47 So we need to translate it into the years into month.

00:52 Now the formula for the average rate of change is s of 12 minus s of 0, divide by 12 minus 0.

01:09 And now this equals to now let's calculate.

01:13 S of 12 s of 0 s of 0 equals to 200 multiplied by 5 minus 9 halves and this could be simplified to 1 ,000 minus and 200 over 2 equals to 100 times 9 is going to be 900 right so this equals to 1 ,000 minus 900 equals to 100 this is a number of units sold at the very beginning now during the first month after 12 months the number of unit salt is going to be 200 times 5 minus 9 over 2 plus 12 is going to be 9 over 14 and this simplifies to 1000 minus well 200 over 14 equal it simplifies to 100 over 7 so it's going to be minus 900 over 7 now let's calculate the now we need to plug in these numbers into the formula.

02:44 We need to plug in s of 12, which is 1000 minus 900 over 7, and subtract s of 0, which equals to 100, and divide this by 12.

03:08 This equals to 900 minus 900 over 7, divided by 12.

03:20 We can factor out 900.

03:22 We are going to get 900 times 1 minus 1 over 7 which equals to 6 over 7 so we are going to get 900 times 6 over 7 divide by 12 this equals to 900 times 6 over 7 times 7 then this simplifies to 900 over 2 times 7 equals to 900 over over 14 which simplifies to 450 over 7.

04:07 And 450 is not divisible by 7.

04:10 So we are going to leave our answer in this form.

04:13 This is an average rate of change during the first year.

04:18 And the units are in units per month...

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