Consecutive even integers are 2 apart
they are
x and x+2
sum is 110
x+x+2=110
2x+2=110
minsu 2 both sides
2x=108
divide both sidedse by 2
x=54
x+2=56
the numbers are 54 and 56
Answer:



Step-by-step explanation:
<u>Optimizing With Derivatives
</u>
The procedure to optimize a function (find its maximum or minimum) consists in
:
- Produce a function which depends on only one variable
- Compute the first derivative and set it equal to 0
- Find the values for the variable, called critical points
- Compute the second derivative
- Evaluate the second derivative in the critical points. If it results positive, the critical point is a minimum, if it's negative, the critical point is a maximum
We know a cylinder has a volume of 4
. The volume of a cylinder is given by

Equating it to 4

Let's solve for h

A cylinder with an open-top has only one circle as the shape of the lid and has a lateral area computed as a rectangle of height h and base equal to the length of a circle. Thus, the total area of the material to make the cylinder is

Replacing the formula of h

Simplifying

We have the function of the area in terms of one variable. Now we compute the first derivative and equal it to zero

Rearranging

Solving for r

![\displaystyle r=\sqrt[3]{\frac{4}{\pi }}\approx 1.084\ feet](https://tex.z-dn.net/?f=%5Cdisplaystyle%20r%3D%5Csqrt%5B3%5D%7B%5Cfrac%7B4%7D%7B%5Cpi%20%7D%7D%5Capprox%201.084%5C%20feet)
Computing h

We can see the height and the radius are of the same size. We check if the critical point is a maximum or a minimum by computing the second derivative

We can see it will be always positive regardless of the value of r (assumed positive too), so the critical point is a minimum.
The minimum area is


the equilibrium point, is when Demand = Supply, namely, when the amount of "Q"uantity demanded by customers is the same as the Quantity supplied by vendors.
That occurs when both of these equations are equal to each other.
let's do away with the denominators, by multiplying both sides by the LCD of all fractions, in this case, 12.
![\bf \stackrel{\textit{Supply}}{-\cfrac{3}{4}Q+35}~~=~~\stackrel{\textit{Demand}}{\cfrac{2}{3}Q+1}\implies \stackrel{\textit{multiplying by 12}}{12\left( -\cfrac{3}{4}Q+35 \right)=12\left( \cfrac{2}{3}Q+1 \right)} \\\\\\ -9Q+420=8Q+12\implies 408=17Q\implies \cfrac{408}{17}=Q\implies \boxed{24=Q} \\\\\\ \stackrel{\textit{using the found Q in the Demand equation}}{P=\cfrac{2}{3}(24)+1}\implies P=16+1\implies \boxed{P=17} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill \stackrel{Equilibrium}{(24,17)}~\hfill](https://tex.z-dn.net/?f=%5Cbf%20%5Cstackrel%7B%5Ctextit%7BSupply%7D%7D%7B-%5Ccfrac%7B3%7D%7B4%7DQ%2B35%7D~~%3D~~%5Cstackrel%7B%5Ctextit%7BDemand%7D%7D%7B%5Ccfrac%7B2%7D%7B3%7DQ%2B1%7D%5Cimplies%20%5Cstackrel%7B%5Ctextit%7Bmultiplying%20by%2012%7D%7D%7B12%5Cleft%28%20-%5Ccfrac%7B3%7D%7B4%7DQ%2B35%20%5Cright%29%3D12%5Cleft%28%20%5Ccfrac%7B2%7D%7B3%7DQ%2B1%20%5Cright%29%7D%20%5C%5C%5C%5C%5C%5C%20-9Q%2B420%3D8Q%2B12%5Cimplies%20408%3D17Q%5Cimplies%20%5Ccfrac%7B408%7D%7B17%7D%3DQ%5Cimplies%20%5Cboxed%7B24%3DQ%7D%20%5C%5C%5C%5C%5C%5C%20%5Cstackrel%7B%5Ctextit%7Busing%20the%20found%20Q%20in%20the%20Demand%20equation%7D%7D%7BP%3D%5Ccfrac%7B2%7D%7B3%7D%2824%29%2B1%7D%5Cimplies%20P%3D16%2B1%5Cimplies%20%5Cboxed%7BP%3D17%7D%20%5C%5C%5C%5C%5B-0.35em%5D%20%5Crule%7B34em%7D%7B0.25pt%7D%5C%5C%5C%5C%20~%5Chfill%20%5Cstackrel%7BEquilibrium%7D%7B%2824%2C17%29%7D~%5Chfill)