Let m be an integer in the set {0,1,2,3,4,5,6,7,8, 9}, and consider the following problem: determine m by asking 3-way questions
, i.e. questions with at most 3 possible answers. For instance, one could ask which of 3 specific subsets m belongs to.
Give a decision tree argument showing that at least 3 such questions are necessary in worst case. In other words, prove that no correct algorithm can solve this problem by asking only 2 questions in worst case.
Give a decision tree argument showing that at least 3 such questions are necessary in worst case. In other words, prove that no correct algorithm can solve this problem by asking only 2 questions in worst case.
1 answer:
Answer:
Take any algorithm if that algorithm solves this problem it can be represented as a ternary decision tree. Therefore each question has at most three answers.
There are ten possible verdicts, the height of such kind of tree should satisfy
ℎ >= ⌈log3(10)⌉ = 3
Hence no such algorithm can ask less than three questions in the worst case.
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b)
Each and every internal node represents a question asking whether m belongs to one of three possible subset of {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} or not
For example 0123|456|789 represented the questionDoes “m: belongs to {0, 1, 2, 3}, to {4, 5, 6}, or to {7, 8, 9}?"
Verdicts are placed in brackets "[ ]"
Explanation:
decision tree is attached below
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Answer:
a) Mechanical efficiency ()=63.15% b) Temperature rise= 0.028ºC
Explanation:
For the item a) you have to define the mechanical power introduced (Wmec) to the system and the power transferred to the water (Pw).
The power input (electric motor) is equal to the motor power multiplied by the efficiency. Thus, .
Then, the power transferred (Pw) to the fluid is equal to the flow rate (Q) multiplied by the pressure jump . So
.
The efficiency is defined as the ratio between the output energy and the input energy. Then, the mechanical efficiency is
For the b) item you have to consider that the inefficiency goes to the fluid as heat. So it is necessary to use the equation of the heat capacity but in a "flux" way. Calling <em>H</em> to the heat transfered to the fluid, the specif heat of the water and the density of the water:
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Finally, the temperature rise is:
Answer:
For aluminum 110.53 C
For copper 110.32 C
Explanation:
Heat transmission through a plate (considering it as an infinite plate, as in omitting the effects at the borders) follows this equation:
Where
q: heat transferred
k: conduction coeficient
A: surface area
th: hot temperature
tc: cold temperature
d: thickness of the plate
Rearranging the terms:
d * q = k * A * (th - tc)
The surface area is:
If the pan is aluminum:
If the pan is copper: