The summation of the considered expression in terms of n from n = 1 to 14 is given by: Option D: 343
<h3>How to find the sum of consecutive integers?</h3>
<h3>
What are the properties of summation?</h3>
![\sum_{i=r}^s (a \times f(i) + b) = a \times [\: \sum_{i=r}^s f(i)] + (s-r)b](https://tex.z-dn.net/?f=%5Csum_%7Bi%3Dr%7D%5Es%20%20%28a%20%5Ctimes%20f%28i%29%20%2B%20b%29%20%3D%20a%20%5Ctimes%20%5B%5C%3A%20%5Csum_%7Bi%3Dr%7D%5Es%20f%28i%29%5D%20%2B%20%28s-r%29b)
where a, b, r, and s are constants, f(i) is function of i, i ranging from r to s (integral assuming).
For the given case, the considered summation can be written symbolically as:
It is evaluated as;
![\sum_{n=1}^{14} (3n + 2) = 3 \times [ \: \sum_{n=1}^{14} n ] + \sum_{n=1}^{14} 2\\\\\sum_{n=1}^{14} (3n + 2) = 3 \times \dfrac{(14)(14 + 1)}{2} + (2 + 2 + .. + 2(\text{14 times}))\\\\\sum_{n=1}^{14} (3n + 2) = 3 \times 105 + 28 = 343\\](https://tex.z-dn.net/?f=%5Csum_%7Bn%3D1%7D%5E%7B14%7D%20%20%283n%20%2B%202%29%20%3D%203%20%5Ctimes%20%5B%20%5C%3A%20%5Csum_%7Bn%3D1%7D%5E%7B14%7D%20n%20%5D%20%2B%20%5Csum_%7Bn%3D1%7D%5E%7B14%7D%202%5C%5C%5C%5C%5Csum_%7Bn%3D1%7D%5E%7B14%7D%20%20%283n%20%2B%202%29%20%3D%203%20%5Ctimes%20%5Cdfrac%7B%2814%29%2814%20%2B%201%29%7D%7B2%7D%20%2B%20%282%20%2B%202%20%2B%20..%20%2B%202%28%5Ctext%7B14%20times%7D%29%29%5C%5C%5C%5C%5Csum_%7Bn%3D1%7D%5E%7B14%7D%20%20%283n%20%2B%202%29%20%3D%203%20%5Ctimes%20105%20%2B%2028%20%3D%20343%5C%5C)
Thus, the summation of the considered expression in terms of n from n = 1 to 14 is given by: Option D: 343
Learn more about summation here:
brainly.com/question/14322177
Until the concerns I raised in the comments are resolved, you can still set up the differential equation that gives the amount of salt within the tank over time. Call it
.
Then the ODE representing the change in the amount of salt over time is
and this with the initial condition
You have
![\dfrac{\mathrm d}{\mathrm dt}\left[e^{t/250}A(t)\right]=\dfrac25e^{t/250}(1+\cos t)](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dt%7D%5Cleft%5Be%5E%7Bt%2F250%7DA%28t%29%5Cright%5D%3D%5Cdfrac25e%5E%7Bt%2F250%7D%281%2B%5Ccos%20t%29)
Integrating both sides gives
Since
, you get
so the amount of salt at any given time in the tank is
The tank will never overflow, since the same amount of solution flows into the tank as it does out of the tank, so with the given conditions it's not possible to answer the question.
However, you can make some observations about end behavior. As
, the exponential term vanishes and the amount of salt in the tank will oscillate between a maximum of about 100.4 lbs and a minimum of 99.6 lbs.
The length that can be cut off from the three given pieces of wood equally to form a right triangle is 5 cm
Given parameters:
dimension of the three pieces of wood = 20 cm, 41 cm and 44 cm
To find:
- the length that is cut off to form a right triangle
let the length that is cut off from each of the wood = y
From Pythagoras theorem, we will have the following equation.
Since the least measurement of one of the pieces of the wood is 20 cm, we cannot cut off 29 cm.
Thus, the highest amount we can cut off equally is 5 cm
learn more here: brainly.com/question/15808950
Answer:
what do u want please say i will try my best to answer it
Answer:
The answer is in the picture I attached.
Step-by-step explanation:
Also, if you need this:
Slope: 1
Y Intercept: (0,6)
Hope this helped!
