The rules of exponents say the exponent outside parentheses applies to each factor inside parentheses.
... (8.1*10^-4)^2 = 8.1^2 × (10^-4)^2
... = 65.61 × 10^-8
... = 6.561 × 10^-7 . . . . adjust to scientific notation with one digit left of the decimal point
The exponent of -7 means the decimal point in the decimal number is 7 places to the left of where it is in scientific notation. That is ...
... 6.561 × 10^-7 = 0.0000006561
Quantity of coffee used per day by breakfast cafe =
Quantity of coffee it will use in a seven day week =
Therefore , a breakfast cafe will use =
By multiplying the daily coffee consumption of 3 1/4 lbs by 7, we find that the cafe uses 22 3/4 lbs of coffee in a week.
In a week, the cafe's coffee consumption is calculated by multiplying its daily usage of 3 1/4 lbs by 7 (the number of days in a week). This computation yields 22 3/4 lbs as the total weekly coffee consumption. This process involves multiplying the daily quantity by the number of days to obtain the weekly total. Therefore, the cafe utilizes 22 3/4 lbs of coffee throughout a 7-day week. This method of determining weekly coffee usage is crucial for managing inventory and ensuring that the cafe meets the demands of its customers efficiently. By understanding their weekly consumption, the cafe can effectively plan and stock their coffee supply.
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Answer with Step-by-step explanation:
m<A=m<J=20
In JOE,
80+20+m<O=180[Angles of a trinagle]
m<O=80=m<M
m<E=m<Y=80
Answer:
1/2
Step-by-step explanation:
Answer:
1/2
Step-by-step explanation:
1/4 = 0.25 x 2 = 0.5 = 1/2
Answer:
Step-by-step explanation:
(a) If the state space is taken as , the probability of transitioning from one state, say (XY) to another state, say (XZ) will be the same as the probability of Y losing out to X, because if X and Y were playing and Y loses to X, then X and Z will play in the next match. This probability is constant with time, as mentioned in the question. Hence, the probabilities of moving from one state to another are constant over time. Hence, the Markov chain is time-homogeneous.
(b) The state transition matrix will be:
where as stated in part (b) above, the rows of the matrix state the probability of transitioning from one of the states (in that order) at time n and the columns of the matrix state the probability of transitioning to one of the states (in the same order) at time n+1.
Consider the entries in the matrix. For example, if players X and Y are playing at time n (row 1), then X beats Y with probability , then since Y is the loser, he sits out and X plays with Z (column 2) at the next time step. Hence, P(1, 2) = . P(1, 1) = 0 because if X and Y are playing, one of them will be a loser and thus X and Y both together will not play at the next time step. , because if X and Y are playing, and Y beats X, the probability of which is, then Y and Z play each other at the next time step. Similarly,, because if X and Z are playing and X beats Z with probability, then X plays Y at the next time step.
(c) At equilibrium,
i.e., the steady state distribution v of the Markov Chain is such that after applying the transition probabilities (i.e., multiplying by the matrix P), we get back the same steady state distribution v. The Eigenvalues of the matrix P are found below:
The solutions are
These are the eigenvalues of P.
The sum of all the rows of the matrix is equal to 0 when Hence, one of the eigenvectors is :
The other eigenvectors can be found using Gaussian elimination:
Hence, we can write:
, where
and
After n time steps, the distribution of states is:
Let n be very large, say n = 1000 (steady state) and let v0 = [0.333 0.333 0.333] be the initial state. then,
Hence,
Now, it can be verified that