Answer:26 first class stamps and 28 additional.
Step-by-step explanation:
From information given,we can formulate the following equation: 47n +25(54-n ) = 1922
Now we solve for n .
47 + 1350+ 25n =1922
47n -25n =1922- 1350
22n=572
n= 26 .
Therefore 26 first class stamps and 28 additional. 54 total
Determine the area of the parallelogram with vertices (4, 6, 2), (4, 7, 2), (9, 7, 2), and (9, 8, 2). Use the square root symbol
astra-53 [7]
Answer:
5 square units
Step-by-step explanation:
The parallelogram lies entirely in the plane z=2, so we can treat this as a 2-dimensional problem.
In the order given, the vertices do not define a parallelogram. Two of the sides are parallel, but the other two sides cross each other, for a net area of zero.
If we swap the order of the last two vertices, we get a parallelogram that has a base of 1 unit and a height of 5 units. Its area is ...
A = bh = 1·5 = 5 . . . square units
Answer:
Part A)
![\displaystyle \frac{dy}{dx}=-\frac{2xy+y^2}{x^2+2xy}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdy%7D%7Bdx%7D%3D-%5Cfrac%7B2xy%2By%5E2%7D%7Bx%5E2%2B2xy%7D)
Part B)
![\displaystyle y=-\frac{5}{8}x+\frac{9}{4}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%3D-%5Cfrac%7B5%7D%7B8%7Dx%2B%5Cfrac%7B9%7D%7B4%7D)
Step-by-step explanation:
We have the equation:
![\displaystyle x^2y+y^2x=6](https://tex.z-dn.net/?f=%5Cdisplaystyle%20x%5E2y%2By%5E2x%3D6)
Part A)
We want to find the derivative of our function, dy/dx.
So, we will take the derivative of both sides with respect to <em>x:</em>
<em />
<em />
The derivative of a constant is 0. We can expand the left:
![\displaystyle \frac{d}{dx}\Big[x^2y\Big]+\frac{d}{dx}\Big[y^2x\Big]=0](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bd%7D%7Bdx%7D%5CBig%5Bx%5E2y%5CBig%5D%2B%5Cfrac%7Bd%7D%7Bdx%7D%5CBig%5By%5E2x%5CBig%5D%3D0)
Differentiate using the product rule:
![\displaystyle \Big(\frac{d}{dx}\big[x^2\big]y+x^2\frac{d}{dx}\big[y\big]\Big)+\Big(\frac{d}{dx}\big[y^2\big]x+y^2\frac{d}{dx}\big[x\big]\Big)=0](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5CBig%28%5Cfrac%7Bd%7D%7Bdx%7D%5Cbig%5Bx%5E2%5Cbig%5Dy%2Bx%5E2%5Cfrac%7Bd%7D%7Bdx%7D%5Cbig%5By%5Cbig%5D%5CBig%29%2B%5CBig%28%5Cfrac%7Bd%7D%7Bdx%7D%5Cbig%5By%5E2%5Cbig%5Dx%2By%5E2%5Cfrac%7Bd%7D%7Bdx%7D%5Cbig%5Bx%5Cbig%5D%5CBig%29%3D0)
Implicitly differentiate:
![\displaystyle (2xy+x^2\frac{dy}{dx})+(2y\frac{dy}{dx}x+y^2)=0](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%282xy%2Bx%5E2%5Cfrac%7Bdy%7D%7Bdx%7D%29%2B%282y%5Cfrac%7Bdy%7D%7Bdx%7Dx%2By%5E2%29%3D0)
Rearrange:
![\displaystyle \Big(x^2\frac{dy}{dx}+2xy\frac{dy}{dx}\Big)+(2xy+y^2)=0](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5CBig%28x%5E2%5Cfrac%7Bdy%7D%7Bdx%7D%2B2xy%5Cfrac%7Bdy%7D%7Bdx%7D%5CBig%29%2B%282xy%2By%5E2%29%3D0)
Isolate the dy/dx:
![\displaystyle \frac{dy}{dx}(x^2+2xy)=-(2xy+y^2)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdy%7D%7Bdx%7D%28x%5E2%2B2xy%29%3D-%282xy%2By%5E2%29)
Hence, our derivative is:
![\displaystyle \frac{dy}{dx}=-\frac{2xy+y^2}{x^2+2xy}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdy%7D%7Bdx%7D%3D-%5Cfrac%7B2xy%2By%5E2%7D%7Bx%5E2%2B2xy%7D)
Part B)
We want to find the equation of the tangent line at (2, 1).
So, let's find the slope of the tangent line using the derivative. Substitute:
![\displaystyle \frac{dy}{dx}_{(2,1)}=-\frac{2(2)(1)+(1)^2}{(2)^2+2(2)(1)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdy%7D%7Bdx%7D_%7B%282%2C1%29%7D%3D-%5Cfrac%7B2%282%29%281%29%2B%281%29%5E2%7D%7B%282%29%5E2%2B2%282%29%281%29%7D)
Evaluate:
![\displaystyle \frac{dy}{dx}_{(2,1)}=-\frac{4+1}{4+4}=-\frac{5}{8}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdy%7D%7Bdx%7D_%7B%282%2C1%29%7D%3D-%5Cfrac%7B4%2B1%7D%7B4%2B4%7D%3D-%5Cfrac%7B5%7D%7B8%7D)
Then by the point-slope form:
![y-y_1=m(x-x_1)](https://tex.z-dn.net/?f=y-y_1%3Dm%28x-x_1%29)
Yields:
![\displaystyle y-1=-\frac{5}{8}(x-2)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y-1%3D-%5Cfrac%7B5%7D%7B8%7D%28x-2%29)
Distribute:
![\displaystyle y-1=-\frac{5}{8}x+\frac{5}{4}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y-1%3D-%5Cfrac%7B5%7D%7B8%7Dx%2B%5Cfrac%7B5%7D%7B4%7D)
Hence, our equation is:
![\displaystyle y=-\frac{5}{8}x+\frac{9}{4}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%3D-%5Cfrac%7B5%7D%7B8%7Dx%2B%5Cfrac%7B9%7D%7B4%7D)