# A factory produces bicycles at a rate of 80+0.5t^2-0.7t bicycles per week (t in weeks). How many bicycles were produced from day 15 to 28?

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Mar 12, 2016

$165$ bicycles

#### Explanation:

The function production is given as
$f \left(t\right) = 80 + 0.5 {t}^{2} - 0.7 t$, $t$ in weeks

The interval to be considered is from day 15 to 28. Or from the third (included) to the fourth week.
For integration purposes it is from the second (it marks the limit of the region to be excluded ) to the fourth week.

So the result is given by

${\int}_{2}^{4} \left(80 + {t}^{2} / 2 - 0.7 t\right) \mathrm{dt} =$
$= \left(80 t + {t}^{3} / 6 - 0.35 {t}^{2}\right) {|}_{2}^{4}$
$= \left(320 + \frac{64}{6} - 0.35 \cdot 16\right) - \left(160 + \frac{8}{6} - 0.35 \cdot 4\right)$
$= 320 + \frac{32}{3} - 5.6 - 160 - \frac{4}{3} + 1.4$
$= 160 + \frac{28}{3} - 4.2 \cong 165.13$ => $165$ bicycles