Integrals Problem

Question

A factory produces bicycles at a rate of 95+0.1t^2 −1.1t bicycles per week (t in weeks). How many bicycles were produced from day 15 to 28?

Answer 1

How many bicycles were produced from day 15 to 28?For integration purposes it is from the second to the fourth week.

integral (95+0.1t^2 −1.1t) dt from , from 2 to 4

= 95x+0.03x^3-(1.1x^2)/2 evaluate from 2 to 4

= 185.26 or 185 bicycles

Answer 2

Hello Dianz!

Since this problem involves the usage of an integral, let's go ahead and set one up. For simplicity, I'm going to use the word integral to depict an integral sign (which remember, looks like an "s" shape).

Let's evaluate what the question is asking for: they are asking you to find how many bicycles are produced from day 15 to 28 (very important to pay attention to the unit here). Notice in the problem, they give you a rate at which the bicycles are produced (remember rate is a derivative-- think rate of change = slope = 1st derivative). Now how do we go from a rate to finding how many bicycles are produced? Simply by taking the antiderivative!

So now that we have a general idea of what to do, let's go back to talking about the units. They wanted us to find how many bicycles are produced from day 15 to 28, but the rate gives us in terms of weeks. This is not a problem because we can easily convert this from 15 days to 2 weeks and 28 days to around 4 weeks.

Alright, now we can finally set up our integral (again, I will use the word integral to depict an integral sign):

integral(from 2 to 4 weeks) of 95+0.1t^2-1.1t dt.

You can solve this by using the FTC or plugging into your calculator.

From this you should get an answer of 185.267, but obviously you round to 185 (because we didn't necessarily reach 186 bicycles) to get 185 bicycles.

Hope that helped!