How many bicycles to the nearest whole number

Question

A factory produces bicycles at a rate of 105+0.4t²-0.9t bicycles per week (in weeks). How many bicycles were produced from day 15 to 28?

I used both days and weeks to find an answer and neither answer was right.

Answer 1

In general, when we're given a rate (such as "bicycles per week") and asked for a quantity (in this case, "bicycles"), our approach will be to integrate the rate with respect to time. We can think about this in terms of units: we're told that the expression 105 + 0.4t² - 0.9t has units of bicycles/week, and the differential dt has units of weeks. When we multiply these things together inside an integral, we find that (105 + 0.4t² - 0.9t)dt has units of (bicycles/week)×(weeks). The weeks cancel, and we're left with units of "bicycles", which is what we want.

The other thing we need to determine is the limits of integration. Week 1 consists of Days 1 through 7 and covers the range 0≤t<1. Week 2 consists of Days 8 through 14 and covers the range 1≤t<2. And so on. Using this, we can determine the range of t values that covers Days 15 through 28 (which is the range they ask us about in the problem) and use these as the start and end points for our integral.

Hope that helps!