Answer:
![y = - \frac{ 1 }{10} {x}^{2} + \frac{5}{2}](https://tex.z-dn.net/?f=y%20%3D%20%20-%20%20%5Cfrac%7B%201%20%7D%7B10%7D%20%7Bx%7D%5E%7B2%7D%20%20%20%2B%20%20%5Cfrac%7B5%7D%7B2%7D%20)
Step-by-step explanation:
We want to prove algebraically that:
![r = \frac{10}{2 + 2 \sin \theta}](https://tex.z-dn.net/?f=r%20%3D%20%20%5Cfrac%7B10%7D%7B2%20%2B%202%20%5Csin%20%5Ctheta%7D%20)
is a parabola.
We use the relations
![{r}^{2} = {x}^{2} + {y}^{2}](https://tex.z-dn.net/?f=%20%7Br%7D%5E%7B2%7D%20%20%3D%20%20%7Bx%7D%5E%7B2%7D%20%20%2B%20%20%7By%7D%5E%7B2%7D%20)
and
![y = r \sin \theta](https://tex.z-dn.net/?f=y%20%3D%20r%20%5Csin%20%5Ctheta)
Before we substitute, let us rewrite the equation to get:
![r(2 + 2 \sin \theta) = 10](https://tex.z-dn.net/?f=r%282%20%2B%202%20%5Csin%20%5Ctheta%29%20%3D%2010)
Or
![r(1+ \sin \theta) = 5](https://tex.z-dn.net/?f=r%281%2B%20%20%5Csin%20%5Ctheta%29%20%3D%205)
Expand :
![r+ r\sin \theta= 5](https://tex.z-dn.net/?f=r%2B%20%20r%5Csin%20%5Ctheta%3D%205)
We now substitute to get:
![\sqrt{ {x}^{2} + {y}^{2} } + y = 5](https://tex.z-dn.net/?f=%20%5Csqrt%7B%20%7Bx%7D%5E%7B2%7D%20%20%2B%20%20%7By%7D%5E%7B2%7D%20%7D%20%20%2B%20y%20%3D%205)
This means that:
![\sqrt{ {x}^{2} + {y}^{2} }=5 - y](https://tex.z-dn.net/?f=%5Csqrt%7B%20%7Bx%7D%5E%7B2%7D%20%20%2B%20%20%7By%7D%5E%7B2%7D%20%7D%3D5%20-%20y)
Square:
![{x}^{2} + {y}^{2} =(5 - y)^{2}](https://tex.z-dn.net/?f=%20%7Bx%7D%5E%7B2%7D%20%20%2B%20%20%7By%7D%5E%7B2%7D%20%3D%285%20-%20y%29%5E%7B2%7D%20)
Expand:
![{x}^{2} + {y}^{2} =25 - 10y + {y}^{2}](https://tex.z-dn.net/?f=%7Bx%7D%5E%7B2%7D%20%20%2B%20%20%7By%7D%5E%7B2%7D%20%3D25%20-%2010y%20%2B%20%20%7By%7D%5E%7B2%7D%20)
![{x}^{2} =25 - 10y](https://tex.z-dn.net/?f=%7Bx%7D%5E%7B2%7D%20%20%3D25%20-%2010y%20)
![{x}^{2} - 25 = - 10y](https://tex.z-dn.net/?f=%7Bx%7D%5E%7B2%7D%20%20-%2025%20%3D%20%20-%2010y%20)
![y = - \frac{ {x}^{2} }{10} + \frac{5}{2}](https://tex.z-dn.net/?f=y%20%3D%20%20-%20%20%5Cfrac%7B%20%7Bx%7D%5E%7B2%7D%20%7D%7B10%7D%20%20%2B%20%20%5Cfrac%7B5%7D%7B2%7D%20)
This is a parabola (0,2.5) and turns upside down.
-2 1/7 is the correct answer.
Answer:
i think it is 1/100
Step-by-step explanation:
4/20^2 add all the marbles number together than put the amount of orange marbles on top of it so it is 4/20 than square it
Answer:
$64.83.
Step-by-step explanation:
Each year the multiplier fffor her new rate will be 1.05.
So after 30 years her hourly rate will be 15 * 1.05^30
= $64.83