Chapter 4 - Integration - 4.5 Exercises - Page 303: 83

$$\eqalign{ & \left( {\text{a}} \right)102.352{\text{ thousands of units}} \cr & \left( {\text{b}} \right)102.352{\text{ thousands of units}} \cr & \left( {\text{c}} \right)74.5{\text{ thousands of units}} \cr} $$

$$\eqalign{ & {\text{The sales }}\left( {{\text{in thousands of units}}} \right){\text{is given by}} \cr & S = 74.50 + 43.75\sin \frac{{\pi t}}{6},{\text{ }}t{\text{ in months}} \cr & {\text{The average value is:}} \cr & {S_{avg}} = \frac{1}{{b - a}}\int_a^b {\left( {74.50 + 43.75\sin \frac{{\pi t}}{6}} \right)dt} \cr & {S_{avg}} = \frac{1}{{b - a}}\left[ {74.50t - \frac{{\left( {43.75} \right)6}}{\pi }\cos \frac{{\pi t}}{6}} \right]_a^b \cr & {S_{avg}} = \frac{1}{{b - a}}\left[ {74.50t - \frac{{262.5}}{\pi }\cos \frac{{\pi t}}{6}} \right]_a^b \cr & \cr & \left( {\text{a}} \right){\text{ The first quarter }}\left( {0 \leqslant t \leqslant 3} \right) \cr & {S_{avg}} = \frac{1}{{3 - 0}}\left[ {74.50t - \frac{{262.5}}{\pi }\cos \frac{{\pi t}}{6}} \right]_0^3 \cr & {S_{avg}} = \frac{1}{3}\left[ {74.50\left( 3 \right) - \frac{{262.5}}{\pi }\cos \frac{{\pi \left( 3 \right)}}{6}} \right] - \frac{1}{3}\left[ { - \frac{{262.5}}{\pi }\cos 0} \right] \cr & {S_{avg}} \approx 102.352{\text{ }} \cr & \cr & \left( {\text{a}} \right){\text{ The first quarter }}\left( {0 \leqslant t \leqslant 3} \right) \cr & {S_{avg}} = \frac{1}{{3 - 0}}\left[ {74.50t - \frac{{262.5}}{\pi }\cos \frac{{\pi t}}{6}} \right]_0^3 \cr & {S_{avg}} = \frac{1}{3}\left[ {74.50\left( 3 \right) - \frac{{262.5}}{\pi }\cos \frac{{\pi \left( 3 \right)}}{6}} \right] - \frac{1}{3}\left[ { - \frac{{262.5}}{\pi }\cos 0} \right] \cr & {S_{avg}} \approx 102.352{\text{ thousands of units}} \cr & \cr & \left( {\text{b}} \right){\text{ The second quarter }}\left( {3 \leqslant t \leqslant 6} \right) \cr & {S_{avg}} = \frac{1}{{6 - 3}}\left[ {74.50t - \frac{{262.5}}{\pi }\cos \frac{{\pi t}}{6}} \right]_3^6 \cr & {S_{avg}} = \frac{1}{3}\left[ {74.50\left( 6 \right) - \frac{{262.5}}{\pi }\cos \frac{{\pi \left( 6 \right)}}{6}} \right] \cr & - \frac{1}{3}\left[ {74.50\left( 3 \right) - \frac{{262.5}}{\pi }\cos \frac{{\pi \left( 3 \right)}}{6}} \right] \cr & {S_{avg}} \approx 102.352{\text{ thousands of units}} \cr & \cr & \left( {\text{c}} \right){\text{ The entire year }}\left( {0 \leqslant t \leqslant 12} \right) \cr & {S_{avg}} = \frac{1}{{12 - 0}}\left[ {74.50t - \frac{{262.5}}{\pi }\cos \frac{{\pi t}}{6}} \right]_0^{12} \cr & {S_{avg}} = \frac{1}{3}\left[ {74.50\left( {12} \right) - \frac{{262.5}}{\pi }\cos \frac{{\pi \left( {12} \right)}}{6}} \right] - \frac{1}{3}\left[ { - \frac{{262.5}}{\pi }} \right] \cr & {S_{avg}} \approx 74.5{\text{ thousands of units}} \cr} $$