This looks like an exercise that's building toward the idea of a derivative.
These calculations are done best with a calculator, but here's how the first interval is used:
Average velocity = (position at 2 - position at 1) / (2 - 1) It's really distance divided by time!
Position at t = 2:
![4\sin{(2\pi)}+3\cos{(2\pi)}=4(0)+3(1)=3](https://tex.z-dn.net/?f=4%5Csin%7B%282%5Cpi%29%7D%2B3%5Ccos%7B%282%5Cpi%29%7D%3D4%280%29%2B3%281%29%3D3)
Position at t = 1:
![4\sin{(\pi)}+3\cos{(\pi)}=4(0)+3(-1)=-3](https://tex.z-dn.net/?f=4%5Csin%7B%28%5Cpi%29%7D%2B3%5Ccos%7B%28%5Cpi%29%7D%3D4%280%29%2B3%28-1%29%3D-3)
So over the interval [1, 2] the average velocity is
![\frac{3-(-3)}{2-1} = \frac{6}{1} = 6](https://tex.z-dn.net/?f=%5Cfrac%7B3-%28-3%29%7D%7B2-1%7D%20%3D%20%5Cfrac%7B6%7D%7B1%7D%20%3D%206)
I used a spreadsheet to calculate the average velocity over the other intervals and a couple of shorter ones, too. (See attached image.)
As these intervals get shorter (the right endpoint is approaching 1), the average velocity gets closer and closer to the instantaneous velocity. An estimate would be -12.6.
Answer:
25.
Step-by-step explanation:
Looks like you are 25 because the grouping is
4 4 4 4 4 4 1
Six sets of 4 and 1 is left over:- 6 * 4 = 24 and plus 1 gives 25.
Total volume of Juice
25 * 12
300 ounces
300 ounces / 32
9.375 quart
9.375 quart / 4
2.34375 gallons
So Earl should buy at least 3 bottles