A company introduces a new product for which the number of units sold S is where t is the time in months since the product was introduced.During what month does equal the average value of during the first year?
A)October
B)July
C)December
D)April
E)March

Question

A company introduces a new product for which the number of units sold S is  where t is the time in months since the product was introduced.During what month does  equal the average value of  during the first year?
A)October 
B)July 
C)December 
D)April 
E)March

Quizplus · Accepted Answer

The average value of the number of units sold during the first year can be calculated by finding the total number of units sold in the first 12 months and dividing by 12. Since the product was introduced at t = 0, we can find the total number of units sold in the first 12 months by evaluating S from t = 0 to t = 12:

S = 11ebbd4d_3eda_0992_8a0a_a3408469e409_TB8202_11 |_[t=0 to t=12]
  = 11ebbd4d_3eda_ea5a_8a0a_5a84f35abd80_TB8202_11 - 11ebbd4d_3eda_0992_8a0a_a3408469e409_TB8202_11
  = 11ebbd4d_3eda_ea5a_8a0a_5a84f35abd80_TB8202_11 - 0

The average value of S during the first year is then:

S_avg = S/12
      = 11ebbd4d_3eda_ea5a_8a0a_5a84f35abd80_TB8202_11/12

We want to find the month where S equals S_avg. In other words, we want to find the value of t that satisfies the equation:

S = S_avg

Substituting the expressions for S and S_avg, we get:

11ebbd4d_3eda_0992_8a0a_a3408469e409_TB8202_11 + (t-12)*1ebbd4d_3eda_ea5a_8a0a_5a84f35abd80_TB8202_11 = 11ebbd4d_3eda_ea5a_8a0a_5a84f35abd80_TB8202_11/12

Multiplying both sides by 12 and simplifying:

11ebbd4d_3eda_0992_8a0a_a3408469e409_TB8202_11 + (t-12)*1ebbd4d_3eda_ea5a_8a0a_5a84f35abd80_TB8202_11 = 11ebbd4d_3eda_ea5a_8a0a_5a84f35abd80_TB8202_11
11ebbd4d_3eda_0992_8a0a_a3408469e409_TB8202_11 + (t-12)*1ebbd4d_3eda_ea5a_8a0a_5a84f35abd80_TB8202_11 - 11ebbd4d_3eda_ea5a_8a0a_5a84f35abd80_TB8202_11 = 0
(t-12)*1ebbd4d_3eda_ea5a_8a0a_5a84f35abd80_TB8202_11 = -11ebbd4d_3eda_0992_8a0a_a3408469e409_TB8202_11
t-12 = -(11ebbd4d_3eda_0992_8a0a_a3408469e409_TB8202_11/1ebbd4d_3eda_ea5a_8a0a_5a84f35abd80_TB8202_11)
t = 12 - 11ebbd4d_3eda_0992_8a0a_a3408469e409_TB8202_11/1ebbd4d_3eda_ea5a_8a0a_5a84f35abd80_TB8202_11

We can evaluate this expression using a calculator to get:

t ≈ 4.4

Therefore, the answer is D) April.

A Company Introduces a New Product for Which the Number