A company introduces a new product for which the number of units sold Ss given by the equation below, where t is the time in months.

s(t) = 155(7-(9/(2+t)))

a) Find he average rate of change of s(t) during the first year.
I got the answer to be 1395/28

b) During what month of the first year does s'(t) equal the average rate of change?
I checked my work and kept getting ~3, which is March, but website is still counting it wrong. Any ideas??

Any help is greatly appreciated!

I agree with the average rate of change. So, you want

s'(c) = 1395/28

s'(t) = 1395/(t+2)^2

so,

1395/(c+2)^2 = 1395/28
(c+2)^2 = 28
c = √28 - 2 ≈ 3.29

So, the desired month is later than 3, namely 4.

So basically round up this time, wow!

To find the average rate of change of s(t) during the first year, you need to evaluate s(t) at the endpoints of the first year and then calculate the difference between the two values divided by the time interval.

a) The first year can be represented by the time period from t = 0 months to t = 12 months. So, you need to find s(0) and s(12) and then calculate (s(12) - s(0)) / (12 - 0).

Let's find the values of s(0) and s(12) first:
s(0) = 155(7 - (9 / (2 + 0)))
= 155(7 - (9 / 2))
= 155(7 - 4.5)
= 155(2.5)
= 387.5

s(12) = 155(7 - (9 / (2 + 12)))
= 155(7 - (9 / 14))
= 155(7 - 0.642857)
= 155(6.357143)
= 985.287

Now, calculating the average rate of change:
average rate of change = (s(12) - s(0)) / (12 - 0)
= (985.287 - 387.5) / (12 - 0)
= 597.787 / 12
= 1395 / 28
= 49.821

So, the average rate of change of s(t) during the first year is 49.821.

b) To find the month during the first year when s'(t) (the derivative of s(t)) equals the average rate of change, you need to find the value of t for which s'(t) is equal to 49.821.

Taking the derivative of s(t):
s'(t) = 155 * (0 - (-9 / (2 + t))^2) * (-1 / (2 + t)^2)
= 155 * (9 / (2 + t))^2 * (1 / (2 + t)^2)
= 155 * (9^2) / (2 + t)^2 * (1 / (2 + t)^2)
= (135 * 9^2) / (2 + t)^4

Now, we need to solve the equation (135 * 9^2) / (2 + t)^4 = 49.821

(135 * 9^2) / (2 + t)^4 = 49.821
Simplifying:
(135 * 81) / (2 + t)^4 = 49.821
(10935) / (2 + t)^4 = 49.821

Now, find the fourth root:
[(10935) / (2 + t)^4]^(1/4) = (49.821)^(1/4)
(10935)^(1/4) / (2 + t) = (49.821)^(1/4)
(10935)^(1/4) = (49.821)^(1/4) * (2 + t)
(10935)^(1/4) - (49.821)^(1/4) = (49.821)^(1/4) * t
t = [(10935)^(1/4) - (49.821)^(1/4)] / (49.821)^(1/4)

Using a calculator, let's calculate t:
t = [18.428 - 3.538] / 3.538
t ≈ 3.3194 / 3.538
t ≈ 0.9396 (rounded off to four decimal places)

Thus, during the first year, s'(t) equals the average rate of change at approximately t = 0.9396 months, which is around the end of January.