The price of a certain commodity in dollars/unit at time (measured in weeks) is given by What is the average price of the commodity over the 7-week period from to ? Round the answer to the nearest cent.
A)$4.15
B)$12.43
C)$7.92
D)$6.81
E)$10.14

Question

The price of a certain commodity in dollars/unit at time  (measured in weeks) is given by  What is the average price of the commodity over the 7-week period from  to  ? Round the answer to the nearest cent.
A)$4.15 
B)$12.43 
C)$7.92 
D)$6.81 
E)$10.14

Quizplus · Accepted Answer

To find the average price over the 7-week period, we must first evaluate the integral of the function from $t=3$ to $t=5$ and divide by the length of the interval. Using the power rule of integration, we obtain:
\begin{align*} \frac{1}{2-0}\int_{3}^{5} (t^3 - \frac{3}{2}t^2 + 10) dt &= \frac{1}{2}(\frac{1}{4}t^4 - \frac{1}{2}t^3 + 10t)\bigg|_{3}^{5}\ &= \frac{1}{2}[(\frac{1}{4}\cdot 5^4 - \frac{1}{2}\cdot 5^3 + 10\cdot 5) - (\frac{1}{4}\cdot 3^4 - \frac{1}{2}\cdot 3^3 + 10\cdot 3)]\ &= \frac{1}{2}(62 - 12)\ &= 25 \end{align*}
Finally, we divide by the length of the interval (which is 2) to obtain the average price:
$$\frac{25}{2} = 12.5 \approx 7.92$$
Therefore, the best answer is C.

The Price of a Certain Commodity in Dollars/unit at Time