Step-by-step explanation:
To determine the future value for the 401(k), we need to determine the amount contributed annually. It is stated that the amount the employee contributes to the fund is 9% of $45,624. $45,624(0.09) = $4,106.16 The employer contributes a maximum of 3% of the employee contribution. Therefore: $45,624(0.03) = $1368.72. Therefore, the total annual contribution is $5,474.88, giving a future value of $196,302.40. For the Roth IRA, the monthly contributions are $352.45 giving a future value of $152,636.09. Therefore, the 401(k) has a greater future value by $43,666.31.
Answer:
x=5/2
y=2
z=-1/2
Step-by-step explanation:
We are given that
![2y+2x=3](https://tex.z-dn.net/?f=2y%2B2x%3D3)
![x+3z=1](https://tex.z-dn.net/?f=x%2B3z%3D1)
![4x-3y+8z=0](https://tex.z-dn.net/?f=4x-3y%2B8z%3D0)
We have to solve the system of equation by using inverse matrix method.
We know that
AX=B
![X=A^{-1} B](https://tex.z-dn.net/?f=X%3DA%5E%7B-1%7D%20B)
Where
A
=![\left[\begin{array}{ccc}0&2&2\\1&0&3\\4&-3&8\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D0%262%262%5C%5C1%260%263%5C%5C4%26-3%268%5Cend%7Barray%7D%5Cright%5D)
![B=\left[\begin{array}{ccc}3\\1\\0\end{array}\right]](https://tex.z-dn.net/?f=B%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D3%5C%5C1%5C%5C0%5Cend%7Barray%7D%5Cright%5D)
![|A|=0(0+9)-2(8-12)+2(-3-0)=2](https://tex.z-dn.net/?f=%7CA%7C%3D0%280%2B9%29-2%288-12%29%2B2%28-3-0%29%3D2)
Now,Minor
,
,![M_{31}=6](https://tex.z-dn.net/?f=M_%7B31%7D%3D6)
,
,![M_{32}=-2](https://tex.z-dn.net/?f=M_%7B32%7D%3D-2)
,
,![M_{33}=-2](https://tex.z-dn.net/?f=M_%7B33%7D%3D-2)
Co-factor
![A_{ij}=(-1)^{i+j}M_{ij}](https://tex.z-dn.net/?f=A_%7Bij%7D%3D%28-1%29%5E%7Bi%2Bj%7DM_%7Bij%7D)
![A_{11}=9,A_{12}=4,A_{13}=-3](https://tex.z-dn.net/?f=A_%7B11%7D%3D9%2CA_%7B12%7D%3D4%2CA_%7B13%7D%3D-3)
![A_{21}=-22,A_{22}=-8,A_{23}=8](https://tex.z-dn.net/?f=A_%7B21%7D%3D-22%2CA_%7B22%7D%3D-8%2CA_%7B23%7D%3D8)
![A_{31}=6,A_{32}=2,A_{33}=-2](https://tex.z-dn.net/?f=A_%7B31%7D%3D6%2CA_%7B32%7D%3D2%2CA_%7B33%7D%3D-2)
Now, adjoint
![adj A=\left[\begin{array}{ccc}9&4&-3\\-22&-8&8\\6&2&-2\end{array}\right]^{T](https://tex.z-dn.net/?f=adj%20A%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D9%264%26-3%5C%5C-22%26-8%268%5C%5C6%262%26-2%5Cend%7Barray%7D%5Cright%5D%5E%7BT)
![adj A=\left[\begin{array}{ccc}9&-22&6\\4&-8&2\\-3&8&-2\end{array}\right]](https://tex.z-dn.net/?f=adj%20A%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D9%26-22%266%5C%5C4%26-8%262%5C%5C-3%268%26-2%5Cend%7Barray%7D%5Cright%5D)
Now,
![A^{-1}=\frac{1}{|A|}adj A](https://tex.z-dn.net/?f=A%5E%7B-1%7D%3D%5Cfrac%7B1%7D%7B%7CA%7C%7Dadj%20A)
![A^{-1}=\frac{1}{2}\left[\begin{array}{ccc}9&-22&6\\4&-8&2\\-3&8&-2\end{array}\right]](https://tex.z-dn.net/?f=A%5E%7B-1%7D%3D%5Cfrac%7B1%7D%7B2%7D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D9%26-22%266%5C%5C4%26-8%262%5C%5C-3%268%26-2%5Cend%7Barray%7D%5Cright%5D)
Now,
![X=\frac{1}{2}\left[\begin{array}{ccc}9&-22&6\\4&-8&2\\-3&8&-2\end{array}\right]\left[\begin{array}{ccc}3\\1\\0\end{array}\right]](https://tex.z-dn.net/?f=X%3D%5Cfrac%7B1%7D%7B2%7D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D9%26-22%266%5C%5C4%26-8%262%5C%5C-3%268%26-2%5Cend%7Barray%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D3%5C%5C1%5C%5C0%5Cend%7Barray%7D%5Cright%5D)
![X=\frac{1}{2}\left[\begin{array}{ccc}27-22\\12-8\\-9+8\end{array}\right]](https://tex.z-dn.net/?f=X%3D%5Cfrac%7B1%7D%7B2%7D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D27-22%5C%5C12-8%5C%5C-9%2B8%5Cend%7Barray%7D%5Cright%5D)
![X=\left[\begin{array}{ccc}5/2\\2\\-1/2\end{array}\right]](https://tex.z-dn.net/?f=X%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D5%2F2%5C%5C2%5C%5C-1%2F2%5Cend%7Barray%7D%5Cright%5D)
![\left[\begin{array}{ccc}x\\y\\z\end{array}\right]=\left[\begin{array}{ccc}5/2\\2\\-1/2\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dx%5C%5Cy%5C%5Cz%5Cend%7Barray%7D%5Cright%5D%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D5%2F2%5C%5C2%5C%5C-1%2F2%5Cend%7Barray%7D%5Cright%5D)
We get
x=5/2
y=2
z=-1/2
Answer:
Step-by-step explanation:
Because of the nature of the information we are given, we have no choice but to use the equation
![y=a(x-h)^2+k](https://tex.z-dn.net/?f=y%3Da%28x-h%29%5E2%2Bk)
and solve for a.
We know by the info that the vertex is (0, 84). We also know that if the vertex is at the origin, and that the base is 42 feet wide, it spans 21 feet to the right of the origin and 21 feet to the left of the origin. That means that we have 2 coordinates from which we need to pick one for our x and y in the equation. I don't like negatives, so I am going to choose the coordinate (21, 0) as x and y. Because this parabola opens upside down, as archways of door openings do, our "a" value better come out algebraically as a negative. Let's see...From the vertex we have that h = 0 and k = 84. So filling in:
and simplifying a bit:
0 = 441a + 84 and
-84 = 441a so
Good, a is negative. Your equation is, then:
![y=-\frac{4}{21}x^2+84](https://tex.z-dn.net/?f=y%3D-%5Cfrac%7B4%7D%7B21%7Dx%5E2%2B84)
Answer:
x= 19
If you need help with more, I'm more than happy to help you.