Question

Company introduces a new product for which the number of units sold S is given by the equation below, where t is the time in months. S(t)=20(5−6+t9 ) (a) Find the average rate of change of S(t) during the first year. (b) During what month of the first year does S(t) equal the average rate of change?

Answer 1

The average rate of change of S(t) during the first year is calculated by finding the change in S(t) divided by the change in time. To find the month in the first year when S(t) equals the average rate of change, we need to set up an equation and solve for t.

Step-by-step explanation:

(a) To find the average rate of change of S(t) during the first year, we need to find the change in S(t) divided by the change in time. We can calculate this by subtracting the initial value of S(t) at t=0 from the final value of S(t) at t=12 (since the first year has 12 months), and then dividing by 12.

(b) To find the month in the first year when S(t) equals the average rate of change, we need to set S(t) equal to the average rate of change and solve for t. To do this, we can set up the equation: 20(5−6+t9) = (S_final - S_initial)/12 and solve for t.

Answer 2

To find the average rate of change of S(t) during the first year, we need to consider the change in the number of units sold over the first 12 months.

(a) To calculate the average rate of change, we need to find the difference in the number of units sold between the initial time (t=0 months) and the end of the first year (t=12 months).

So, let's evaluate S(t) at t=0 months and t=12 months:

S(0) = 20(5 - 6 + 0/9) = 20(5 - 6) = 20(-1) = -20

S(12) = 20(5 - 6 + 12/9) = 20(5 - 6 + 4/3) = 20(5 - 6 + 1.33) = 20 * 0.33 = 6.6

Now, we can calculate the average rate of change:

Average rate of change = (S(12) - S(0)) / (12 - 0)

= (6.6 - (-20)) / 12

= (6.6 + 20) / 12

= 26.6 / 12

= 2.2167 (rounded to 4 decimal places)

Therefore, the average rate of change of S(t) during the first year is approximately 2.2167.

(b) To determine the month during the first year when S(t) equals the average rate of change, we need to solve the equation S(t) = Average rate of change.

Let's set up the equation:

20(5 - 6 + t/9) = 2.2167

Simplifying the equation:

100 - 120 + t/9 = 2.2167

-20 + t/9 = 2.2167

t/9 = 2.2167 + 20

t/9 = 22.2167

t = 22.2167 * 9

t ≈ 199.95

Therefore, during which month of the first year does S(t) equal the average rate of change, approximately?

It seems there is an error in the equation or calculation as the value of t exceeds 12 months. Please double-check the equation or provide additional information for a more accurate analysis.