Formula for Riemann Sum is:
![\frac{b-a}{n} \sum_{i=1}^n f(a + i \frac{b-a}{n})](https://tex.z-dn.net/?f=%5Cfrac%7Bb-a%7D%7Bn%7D%20%5Csum_%7Bi%3D1%7D%5En%20f%28a%20%2B%20i%20%5Cfrac%7Bb-a%7D%7Bn%7D%29)
interval is [1,3] so a = 1, b = 3
f(x) = 3x , sub into Riemann sum
![\frac{2}{n} \sum_{i=1}^n 3(1 + \frac{2i}{n})](https://tex.z-dn.net/?f=%5Cfrac%7B2%7D%7Bn%7D%20%5Csum_%7Bi%3D1%7D%5En%203%281%20%2B%20%5Cfrac%7B2i%7D%7Bn%7D%29)
Continue by simplifying using properties of summations.
![= \frac{2}{n}\sum_{i=1}^n 3 + \frac{2}{n}\sum_{i=1}^n \frac{6i}{n} \\ \\ = \frac{6}{n}\sum_{i=1}^n 1 + \frac{12}{n^2}\sum_{i=1}^n i \\ \\ =\frac{6}{n} (n) + \frac{12}{n^2}(\frac{n(n+1)}{2}) \\ \\ =6+\frac{6}{n}(n+1) \\ \\ =12 + \frac{6}{n}](https://tex.z-dn.net/?f=%3D%20%5Cfrac%7B2%7D%7Bn%7D%5Csum_%7Bi%3D1%7D%5En%203%20%2B%20%20%5Cfrac%7B2%7D%7Bn%7D%5Csum_%7Bi%3D1%7D%5En%20%5Cfrac%7B6i%7D%7Bn%7D%20%5C%5C%20%20%5C%5C%20%3D%20%5Cfrac%7B6%7D%7Bn%7D%5Csum_%7Bi%3D1%7D%5En%201%20%2B%20%20%5Cfrac%7B12%7D%7Bn%5E2%7D%5Csum_%7Bi%3D1%7D%5En%20i%20%5C%5C%20%20%5C%5C%20%3D%5Cfrac%7B6%7D%7Bn%7D%20%28n%29%20%2B%20%5Cfrac%7B12%7D%7Bn%5E2%7D%28%5Cfrac%7Bn%28n%2B1%29%7D%7B2%7D%29%20%5C%5C%20%20%5C%5C%20%3D6%2B%5Cfrac%7B6%7D%7Bn%7D%28n%2B1%29%20%5C%5C%20%20%5C%5C%20%3D12%20%2B%20%5Cfrac%7B6%7D%7Bn%7D%20)
Now you have an expression for the summation in terms of 'n'.
Next, take the limit as n-> infinity.
The limit of
![\frac{6}{n}](https://tex.z-dn.net/?f=%5Cfrac%7B6%7D%7Bn%7D)
goes to 0, therefore the limit of the summation is 12.
The area under the curve from [1,3] is equal to limit of summation which is 12.
Fraction of tank drops in every 10 minutes = 1/3 inches.
We need to find the expected change in the tank's water level after 2.25 hours.
Let us convert 2.25 hours in minutes.
1 hours = 60 minutes.
2.25 hours = 2.25 × 60 = 135 minutes.
<em>We need to divide 135 by 10.</em>
We get 13.5.
Now, we need to <em>multiply 1/3 by 13.5</em>, we get.
13.5/3 = 4.5 inches.
<h3>So, tank would drop 4.5 inches after 2.25 hours.</h3>
6. 6th grade- go to the two columns above $600, which is about 8 students in the first column and 5 students in the 2nd. 8 + 5 = 13, so 13 students in 6th grade earned $600 or more.
7th grade- again, go to the columns above $600, which is about 7 students in the first column and about 3 in the 2nd column. 7 + 3 = 10, so 10 students in 7th grade earned $600 or more.
Do the same thing for #7, just follow the rows exactly as #6.
I repaired the damage from car accidents.
Answer:
slope is y=10x.
Slope shows that she is consistently burning 10 calories per minute
Step-by-step explanation: