w3e_average

w3e_average - t × V t We are only interested in the...

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1 Problem. Averages Consider the plot below in which we have drawn a continuous function of time and evaluated it at the center of 10 intervals. a) Write an expression for the average of these 10 points. b) Generalize this expression to N points and write it in “sigma” notation. c) Multiply and divide the expression above by the interval length Δ t . What is N Δ t equal to? Rewrite the expression using this fact.

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2 d) Take the limit as N goes to inﬁnity. Note that Δ t goes to dt , and the sum becomes an inte- gral. This limit is deﬁned as the average value of a function on the interval [0 , T ] . e) Household voltage is a function of the form V ( t ) = V 0 cos( ωt ) where ω is the radian fre- quency version of 60 Hz, and V 0 = 170 V. Your conventional oven is equivalent to a resistor R . From Ohm’s Law the current drawn is I ( t ) = V ( t ) /R , where R is about 20 Ω . The power used by your oven is measured in watts [ W ] and at any given point in time it is P ( t ) = I (
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Unformatted text preview: t ) × V ( t ) . We are only interested in the average power since this is what the energy company uses to decide how much to charge you. Find the average power used by your oven. (Note that P ( t ) is periodic and you only need to compute an average over its one period.) It is useful to note the trigonometric identity, cos 2 ( x ) = 1 2 ( 1 + cos(2 x ) ) . f) Recall that the total energy used can be computed as (the average power) times (the total time) and that the energy is often measured in kilowatt-hours. If your oven takes 4 hours to cook a turkey and the energy company charges 20 cents/kilowatt-hour, how much does it cost to cook the meal? If you want to learn more about problems of this kind, consider taking the following engi-neering courses: ECE 2100 Introduction to Circuits for Electrical and Computer Engineers (also ENGRD 2100)....
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This note was uploaded on 02/25/2009 for the course MATH 1910 taught by Professor Berman during the Fall '07 term at Cornell.

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w3e_average - t × V t We are only interested in the...

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