43.68. You multiply the base by the height
Recall the following identities:
sin²(<em>x</em>) = (1 - cos(2<em>x</em>))/2
cos²(<em>x</em>) = (1 + cos(2<em>x</em>))/2
Then
sin²(<em>π</em>/8) = (1 - cos(<em>π</em>/4))/2 = (1 - 1/√2)/2
cos⁴(3<em>π</em>/8) = (cos²(3<em>π</em>/8))² = ((1 + cos(3<em>π</em>/4))/2)² = ((1 - 1/√2)/2)²
and so
sin²(<em>π</em>/8) - cos⁴(3<em>π</em>/8) = (1 - 1/√2)/2 - ((1 - 1/√2)/2)²
… = (1 - 1/√2)/2 • (1 - (1 - 1/√2)/2)
… = (2 - √2)/4 • (1 - (2 - √2)/4)
… = (2 - √2)/16 • (4 - 2 + √2)
… = (2 - √2)(2 + √2)/16
… = (4 - 2)/16
… = 1/8
Answer:
The probability of the system being down in the next hour of operation is 0.3.
Step-by-step explanation:
We have a transition matrix from one period to the next (one hour) that can be written as:
![T=\left[\begin{array}{ccc}&R&D\\R&0.7&0.3\\D&0.2&0.8\end{array}\right]](https://tex.z-dn.net/?f=T%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%26R%26D%5C%5CR%260.7%260.3%5C%5CD%260.2%260.8%5Cend%7Barray%7D%5Cright%5D)
We can represent the state that system is initially running with the vector:
![S_0=\left[\begin{array}{cc}1&0\end{array}\right]](https://tex.z-dn.net/?f=S_0%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D1%260%5Cend%7Barray%7D%5Cright%5D)
The probabilties of the states in the next period can be calculated using the matrix product of the actual state and the transition matrix:
That is:
![S_1=S_0\cdot T= \left[\begin{array}{cc}1&0\end{array}\right]\cdot \left[\begin{array}{cc}0.7&0.3\\0.2&0.8\end{array}\right]= \left[\begin{array}{cc}0.7&0.3\end{array}\right]](https://tex.z-dn.net/?f=S_1%3DS_0%5Ccdot%20T%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D1%260%5Cend%7Barray%7D%5Cright%5D%5Ccdot%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D0.7%260.3%5C%5C0.2%260.8%5Cend%7Barray%7D%5Cright%5D%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D0.7%260.3%5Cend%7Barray%7D%5Cright%5D)
With the inital state as running, we have a probabilty of 0.7 that the system will be running in the next hour and a probability of 0.3 that it will be down.
Answer:
n = 13
Step-by-step explanation:
374 = 11(2n+8)
Expand the right side
374 = 11* 2n + 11 * 8
374 = 22n + 88
Subtract 88 from both sides
374 - 88 = 22n
286 = 22n
divide both sides by 22
n = 13
C and D
Why is there a character limit?
