A company introduces a new product for which the number of units sold S is $S ( t ) = 300 \left( 7 - \frac { 10 } { 5 + t } \right)$ where $t$ is the time in months since the product was introduced. Find the average value of $S ( t )$ during the first year.
A) $\frac { 600 } { 17 }$
B) $\frac { 600 } { 11 }$
C) $\frac { 300 } { 17 }$
D) $\frac { 600 } { 17 }$.
E) $\frac { 3000 } { 17 }$.

Question

Quizplus · Accepted Answer

The average value of a function $f(t)$ over the interval $[a, b]$ is given by $\frac{1}{b-a} \int_{a}^{b} f(t) dt$. For $S(t) = 300(7 - \frac{10}{5+t})$ over the first year ($0 \leq t \leq 12$), the average value is $\frac{1}{12-0} \int_{0}^{12} 300(7 - \frac{10}{5+t}) dt$. Simplifying this integral and evaluating gives the average value as $\frac{600}{17}$.

A Company Introduces a New Product for Which the Number S(t)=300(7−105+t)S ( t ) = 300 \left( 7 - \frac { 10 } { 5 + t } \right)S(t)=300(7−5+t10​)

A Company Introduces a New Product for Which the Number $S ( t ) = 300 \left( 7 - \frac { 10 } { 5 + t } \right)$