9514 1404 393
Answer:
4 1/2 days
Step-by-step explanation:
The time is given by ...
time = quantity/rate
time = (3/4 fence)/(1/6 fence/day) = (3/4)(6) days = 4 1/2 days
![\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:
The answers for your two problems, can be found below in the attached images.
Problem 1
Graph
The graph was plotted using a calculator. We can see that the graph opens up.
f(x) = x^2 + 4x +3
Vertex
The vertex is the minimum point of the equation
In this case Vertex (V) = (-2,-1)
Axis of symmetry
The x axis corresponding to the vertex component
x = -2
y intercept
Interception with the y-axis
From the first attached image, we can see that they y intercept occurs
at x = 0, y = 3
(0,3)
Problem 2
Graph.
The graph was plotted using a calculator. We can see that the graph opens up.
f(x) = 2x^2 + 3x +1
Vertex
The vertex is the minimum point of the equation
In this case Vertex (V) = (-0.75,-0.125)
Axis of symmetry
The x axis corresponding to the vertex component
x = -0.75
y intercept
Interception with the y-axis
From the second attached image, we can see that they y intercept occurs
at x = 0, y = 1
(0,1)