Answer:
x=12
Step-by-step explanation:
Step 1: Subtract 7 from both sides.
Step 2: Multiply both sides by 4.
and you get x=12
H(x)=4 because if you make the x next to 4 is 3 so then you solve from there to get 4
![\Bbb E[Y^2] = \Bbb E[(4X - 2)^2]](https://tex.z-dn.net/?f=%5CBbb%20E%5BY%5E2%5D%20%3D%20%5CBbb%20E%5B%284X%20-%202%29%5E2%5D)
so by definition of expectation,
![\Bbb E[Y^2] = \displaystyle \int_{-\infty}^\infty (4x-2)^2 f_X(x) \, dx = \int_0^\infty 3(4x-2)^2 e^{-3x} \, dx](https://tex.z-dn.net/?f=%5CBbb%20E%5BY%5E2%5D%20%3D%20%5Cdisplaystyle%20%5Cint_%7B-%5Cinfty%7D%5E%5Cinfty%20%284x-2%29%5E2%20f_X%28x%29%20%5C%2C%20dx%20%3D%20%5Cint_0%5E%5Cinfty%203%284x-2%29%5E2%20e%5E%7B-3x%7D%20%5C%2C%20dx)
Integrate by parts (twice).
First, let
so that
![\displaystyle \Bbb E[Y^2] = -(4x-2)^2 e^{-3x} \bigg|_{x=0}^{x\to\infty} + 8 \int_0^\infty (4x-2) e^{-3x} \, dx \\\\ ~~~~ = 4 + 8 \int_0^\infty (4x-2) e^{-3x} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5CBbb%20E%5BY%5E2%5D%20%3D%20-%284x-2%29%5E2%20e%5E%7B-3x%7D%20%5Cbigg%7C_%7Bx%3D0%7D%5E%7Bx%5Cto%5Cinfty%7D%20%2B%208%20%5Cint_0%5E%5Cinfty%20%284x-2%29%20e%5E%7B-3x%7D%20%5C%2C%20dx%20%5C%5C%5C%5C%20~~~~%20%3D%204%20%2B%208%20%5Cint_0%5E%5Cinfty%20%284x-2%29%20e%5E%7B-3x%7D%20%5C%2C%20dx)
Next,
so that
![\displaystyle \Bbb E[Y^2] = 4 + 8 \left(-\frac13 (4x-2) e^{-3x} \bigg|_{x=0}^{x\to\infty} + \frac43 \int_0^\infty e^{-3x} \, dx\right) \\\\ ~~~~ = 4 + 8 \left(-\frac23 - \frac49 e^{-3x}\bigg|_{x=0}^{x\to\infty}\right) \\\\ ~~~~ = 4 - \frac{16}3 + \frac{32}9 = \boxed{\frac{20}9}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5CBbb%20E%5BY%5E2%5D%20%3D%204%20%2B%208%20%5Cleft%28-%5Cfrac13%20%284x-2%29%20e%5E%7B-3x%7D%20%5Cbigg%7C_%7Bx%3D0%7D%5E%7Bx%5Cto%5Cinfty%7D%20%2B%20%5Cfrac43%20%5Cint_0%5E%5Cinfty%20e%5E%7B-3x%7D%20%5C%2C%20dx%5Cright%29%20%5C%5C%5C%5C%20~~~~%20%3D%204%20%2B%208%20%5Cleft%28-%5Cfrac23%20-%20%5Cfrac49%20e%5E%7B-3x%7D%5Cbigg%7C_%7Bx%3D0%7D%5E%7Bx%5Cto%5Cinfty%7D%5Cright%29%20%5C%5C%5C%5C%20~~~~%20%3D%204%20-%20%5Cfrac%7B16%7D3%20%2B%20%5Cfrac%7B32%7D9%20%3D%20%5Cboxed%7B%5Cfrac%7B20%7D9%7D)
Answer:
(1 , 6)
Step-by-step explanation:
the center of the circle is the midpoint of the diameter
Which is the point of coordinates:
Then the center is the point :
On the 40th and the 80th you will receive both a free beverage and a free appetizer. You can find this by simply finding the multiples of both number up to 100 (or a little more than 100) to find out on which dates you'd get bothe an appetizer and beverage for free. Here are the multiples of both, to prove this answer.
- 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104.
- 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100.
Thus making 40 and 80 the answers. I hope this helps!
