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Math Help - Component of an integral problem

  1. #1
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    Component of an integral problem

    "Throughout much of the 20th century, the yearly consumption of electricity in the US increased exponentially at a continuous rate of 7% a year. Assume this trend continues and that the electrical energy consumed an 1900 was 1.4 million megawatt hours.

    a)write an expression for the consumption since 1900"

    For this I made the expression C(t)=1.4 X (1.07)^t

    "b)find the average yearly consumption throughout the 20th century"

    for this, I found the amount in the year 2000 by letting t equal 100, added it to the initial and divided it by 2. This yielded me an average of 608.102.

    "c) during what year was the consumption closest to the average?"

    I used logs and such to find the value of t when the above value was substituted into the equation, and found t equaled 89 . However, the program I'm using to check my answers for this assignment says that the years is 1972. Could someone tell me if i'm doing something wrong? or if the professor made an error in entering the answer into the program.
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  2. #2
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    Quote Originally Posted by Nitz456 View Post
    "Throughout much of the 20th century, the yearly consumption of electricity in the US increased exponentially at a continuous rate of 7% a year. Assume this trend continues and that the electrical energy consumed an 1900 was 1.4 million megawatt hours.

    a)write an expression for the consumption since 1900"

    For this I made the expression C(t)=1.4 X (1.07)^t

    "b)find the average yearly consumption throughout the 20th century"

    for this, I found the amount in the year 2000 by letting t equal 100, added it to the initial and divided it by 2. This yielded me an average of 608.102.

    "c) during what year was the consumption closest to the average?"

    I used logs and such to find the value of t when the above value was substituted into the equation, and found t equaled 89 . However, the program I'm using to check my answers for this assignment says that the years is 1972. Could someone tell me if i'm doing something wrong? or if the professor made an error in entering the answer into the program.
    The problem is part b). Evidently this is a class that uses Calculus, so at some point you must have gone over the mean value theorem for integration. If you were finding the average of a linear function you did it the correct way. But your function is an exponential, and the average will be fairly different.

    The general theorem is, given a function f(x) and a domain [a, b] over which you are finding the average:
    \overline{f(x)} = \frac{1}{b - a} \int_a^b f(x) dx

    In this case we want the average of the consumption over the 20th century, so
    \overline{C(t)} = \frac{1}{100} \int_0^{100} 1.4 \cdot (1.07)^t dt

    The integration is fairly trivial and pointing out that \int a^x dx = \frac{a^x}{ln(x)} + C I'll let you do the rest.

    I got that \overline{C(t)} \approx 179.342 which indeed gives t = 72 as an answer for c).

    -Dan
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  3. #3
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    Thank you profusely, I completely forgot about the MVT.
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