Answer:
The completed reaction is
CaCO3 + 2 HCl → (Ca+2 + 2 Cl-)aq + H2O + CO2↑
Answer:
y=x/2-2
Step-by-step explanation:
it's actually 1/2 times x is how your supposed to write it (like a fraction)
So we can call the width of the rectangle x.
So then the length would be 4x.
The perimeter would then be 10x.
Since the perimeter is 70, we can say 70 = 10x.
And then simplify that to 7 = x.
So the length of a rectangle would be 28 cm, and the width would be 7.
So to find the area, just multiply length by width.
28*7 = 196 square cm
So the rectangle's area is 196 square cm.
I hope I helped!
In Cartesian coordinates, the region (call it
) is the set
![R = \left\{(x,y,z) ~:~ x\ge0 \text{ and } y\ge0 \text{ and } 2 \le z \le 4-x^2-y^2\right\}](https://tex.z-dn.net/?f=R%20%3D%20%5Cleft%5C%7B%28x%2Cy%2Cz%29%20~%3A~%20x%5Cge0%20%5Ctext%7B%20and%20%7D%20y%5Cge0%20%5Ctext%7B%20and%20%7D%202%20%5Cle%20z%20%5Cle%204-x%5E2-y%5E2%5Cright%5C%7D)
In the plane
, we have
![2 = 4 - x^2 - y^2 \implies x^2 + y^2 = 2 = \left(\sqrt2\right)^2](https://tex.z-dn.net/?f=2%20%3D%204%20-%20x%5E2%20-%20y%5E2%20%5Cimplies%20x%5E2%20%2B%20y%5E2%20%3D%202%20%3D%20%5Cleft%28%5Csqrt2%5Cright%29%5E2)
which is a circle with radius
. Then we can better describe the solid by
![R = \left\{(x,y,z) ~:~ 0 \le x \le \sqrt2 \text{ and } 0 \le y \le \sqrt{2 - x^2} \text{ and } 2 \le z \le 4 - x^2 - y^2 \right\}](https://tex.z-dn.net/?f=R%20%3D%20%5Cleft%5C%7B%28x%2Cy%2Cz%29%20~%3A~%200%20%5Cle%20x%20%5Cle%20%5Csqrt2%20%5Ctext%7B%20and%20%7D%200%20%5Cle%20y%20%5Cle%20%5Csqrt%7B2%20-%20x%5E2%7D%20%5Ctext%7B%20and%20%7D%202%20%5Cle%20z%20%5Cle%204%20-%20x%5E2%20-%20y%5E2%20%5Cright%5C%7D)
so that the volume is
![\displaystyle \iiint_R dV = \int_0^{\sqrt2} \int_0^{\sqrt{2-x^2}} \int_2^{4-x^2-y^2} dz \, dy \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Ciiint_R%20dV%20%3D%20%5Cint_0%5E%7B%5Csqrt2%7D%20%5Cint_0%5E%7B%5Csqrt%7B2-x%5E2%7D%7D%20%5Cint_2%5E%7B4-x%5E2-y%5E2%7D%20dz%20%5C%2C%20dy%20%5C%2C%20dx)
While doable, it's easier to compute the volume in cylindrical coordinates.
![\begin{cases} x = r \cos(\theta) \\ y = r\sin(\theta) \\ z = \zeta \end{cases} \implies \begin{cases}x^2 + y^2 = r^2 \\ dV = r\,dr\,d\theta\,d\zeta\end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D%20x%20%3D%20r%20%5Ccos%28%5Ctheta%29%20%5C%5C%20y%20%3D%20r%5Csin%28%5Ctheta%29%20%5C%5C%20z%20%3D%20%5Czeta%20%5Cend%7Bcases%7D%20%5Cimplies%20%5Cbegin%7Bcases%7Dx%5E2%20%2B%20y%5E2%20%3D%20r%5E2%20%5C%5C%20dV%20%3D%20r%5C%2Cdr%5C%2Cd%5Ctheta%5C%2Cd%5Czeta%5Cend%7Bcases%7D)
Then we can describe
in cylindrical coordinates by
![R = \left\{(r,\theta,\zeta) ~:~ 0 \le r \le \sqrt2 \text{ and } 0 \le \theta \le\dfrac\pi2 \text{ and } 2 \le \zeta \le 4 - r^2\right\}](https://tex.z-dn.net/?f=R%20%3D%20%5Cleft%5C%7B%28r%2C%5Ctheta%2C%5Czeta%29%20~%3A~%200%20%5Cle%20r%20%5Cle%20%5Csqrt2%20%5Ctext%7B%20and%20%7D%200%20%5Cle%20%5Ctheta%20%5Cle%5Cdfrac%5Cpi2%20%5Ctext%7B%20and%20%7D%202%20%5Cle%20%5Czeta%20%5Cle%204%20-%20r%5E2%5Cright%5C%7D)
so that the volume is
![\displaystyle \iiint_R dV = \int_0^{\pi/2} \int_0^{\sqrt2} \int_2^{4-r^2} r \, d\zeta \, dr \, d\theta \\\\ ~~~~~~~~ = \frac\pi2 \int_0^{\sqrt2} \int_2^{4-r^2} r \, d\zeta\,dr \\\\ ~~~~~~~~ = \frac\pi2 \int_0^{\sqrt2} r((4 - r^2) - 2) \, dr \\\\ ~~~~~~~~ = \frac\pi2 \int_0^{\sqrt2} (2r-r^3) \, dr \\\\ ~~~~~~~~ = \frac\pi2 \left(\left(\sqrt2\right)^2 - \frac{\left(\sqrt2\right)^4}4\right) = \boxed{\frac\pi2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Ciiint_R%20dV%20%3D%20%5Cint_0%5E%7B%5Cpi%2F2%7D%20%5Cint_0%5E%7B%5Csqrt2%7D%20%5Cint_2%5E%7B4-r%5E2%7D%20r%20%5C%2C%20d%5Czeta%20%5C%2C%20dr%20%5C%2C%20d%5Ctheta%20%5C%5C%5C%5C%20~~~~~~~~%20%3D%20%5Cfrac%5Cpi2%20%5Cint_0%5E%7B%5Csqrt2%7D%20%5Cint_2%5E%7B4-r%5E2%7D%20r%20%5C%2C%20d%5Czeta%5C%2Cdr%20%5C%5C%5C%5C%20~~~~~~~~%20%3D%20%5Cfrac%5Cpi2%20%5Cint_0%5E%7B%5Csqrt2%7D%20r%28%284%20-%20r%5E2%29%20-%202%29%20%5C%2C%20dr%20%5C%5C%5C%5C%20~~~~~~~~%20%3D%20%5Cfrac%5Cpi2%20%5Cint_0%5E%7B%5Csqrt2%7D%20%282r-r%5E3%29%20%5C%2C%20dr%20%5C%5C%5C%5C%20~~~~~~~~%20%3D%20%5Cfrac%5Cpi2%20%5Cleft%28%5Cleft%28%5Csqrt2%5Cright%29%5E2%20-%20%5Cfrac%7B%5Cleft%28%5Csqrt2%5Cright%29%5E4%7D4%5Cright%29%20%3D%20%5Cboxed%7B%5Cfrac%5Cpi2%7D)
Answer:
(4x+3) (4x-3)
Step-by-step explanation: