Integration problem: electricity consumption in the US

[Hockeyman]

1) Throughout much of the century, the yearly consumption of electricity in the US has been increasing exponentially at a continous rate of 7% per year. Assuming this trend continues and that the electrical energy consumed in 1900 was 1.4 million mefawatt-hours

Now i found the average energy to be 2.19 x 10^8 megawatt-hours, now part C) asks when is the electrical consumption closest to the average for the century. then part D) asks without calculation how would you have predicted whcih half of the century the answer should be in?

I'm really not sure how to approach these two parts, any help?

 

[skeeter]

Re: Integration problems

1) Throughout much of the century, the yearly consumption of electricity in the US has been increasing exponentially at a continous rate of 7% per year. Assuming this trend continues and that the electrical energy consumed in 1900 was 1.4 million mefawatt-hours

is there information about the problem missing between what you posted above and what you posted below?

Now i found the average energy to be 2.19 x 10^8 megawatt-hours, now part C) asks when is the electrical consumption closest to the average for the century.
then part D) asks without calculation how would you have predicted whcih half of the century the answer should be in?

I'm really not sure how to approach these two parts, any help?

 

[Hockeyman]

Re: Integration problems

well part A) asks to write an expression for yearly electricity consumption as a function of time i found that to be y=1.4x10^8* e^(.07)t, then part B) asks you to find the average yearly consumption which i found to be 2.19 x 10^8

 

[skeeter]

Re: Integration problems

well part A) asks to write an expression for yearly electricity consumption as a function of time i found that to be y=1.4x10^8* e^(.07)t, then part B) asks you to find the average yearly consumption which i found to be 2.19 x 10^8

for part (A) ... one million is 10[sup:2afwwk6h]6[/sup:2afwwk6h] ... how did you get "y=1.4x10^8* e^(.07)t" ?

for part (B) find the average yearly consumption over what period of time?

 

[Hockeyman]

Re: Integration problems

i'm sorry that was an error on my part for part A) i got 1.4x10^6, and for part B) the time periosd is throughout the century

 

[skeeter]

Re: Integration problems

\(\displaystyle y(t) = (1.4 \times 10^6)e^{.07t}\)

average energy consumption for the century is o.k.

\(\displaystyle y_{avg} = \frac{1}{100} \int_0^{100} y(t) dt = 2.19 \times 10^8\) megawatt-hrs/year

for part (c) ... set y(t) = y[sub:1ayshbg9]avg[/sub:1ayshbg9] and solve for t.

for part (d) ... why would that time found in part (c) be in the latter half of the century? think about the rate of consumption ... how is it increasing over the years?

 

[Hockeyman]

for part C) i got t= 72.18 years, is this correct? However i'm still not sure how to answer part D), it increases at a continous rate of 7% every year, but without actually doing calculations i dont see how you would know the answer would be in the second half of the century.

 

[Dr. Flim-Flam]

The rate of change of electricity used in the United States is directly proportional to the amount of electricity used.

Ergo dE/dt = kE, given k = .07, hence dE/dt = .07E implies E(t) = Ae^.07t

Letting 1900 = 0, E(0) = 1.4 = A; henceforth E(t) = 1.4e^.07t (1.4 = 1.4*10^6 mega-watt hrs.).

Average usuage throughout the century = (1/100)integral[1.4e^.07t,t,0,100] = 219 (2.19*10^8 mega-watts hrs).

219 = 1.4e^.07t, t = 72.18 yrs.(about). In the year 1972, the amt of electricity used = about the average used throughout the

century.

Let f(t) = 1.4e^.07t, then limit f(t) as t approaches infinity = infinity,

hence as time goes by the amount of electricty used (according to this equation) goes through the roof.