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Problem 39 Easy Difficulty

Let $ P $ represent the percentage of a city's electrical power that is produced by solar panels $ t $ years after January 1, 2000.

(a) What does $ dP/dt $ represent in this context?
(b) Interpret the statement $$ \frac{dP}{dt} \bigg|_{t = 2} = 3.5 $$

Answer

(a) (a) $d P / d t$ is the rate at which the percentage of the city's electrical power produced by solar panels changes with respect to time $t,$ measured in percentage points per year.
(b) $2 \text { years after January } 1,2000 \text { (January } 1,2002),$ the percentage of electrical power produced by solar panels was increasing at a rate of 3.5 percentage points per year.

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Video Transcript

Okay, P. Is the percentage of power produced um By solar panels and T. is the number of years since the year 2000 uh derivative, the derivative of P with respect 22 Ah What does this mean? This means, what is the rate of change in the percentage of power produced with respect to a change in time? The rate at which the percentage of power produces changes with respect to a change in time. Next. If we have the derivative of P with respect to T evaluated When T. Equals two. Member T. Is the number of years. So two years since 2000, if this value equals 3.5. Ah What does this mean? Well, the derivative of P. With respect to T. Think of that as the change in P. With respect to a change in T. Uh The derivative of P with respect to T. Is actually the limit of this quotation as delta T. Goes to zero, but we can approximate it. The derivative A. P with respect to T. Can be approximated by this quotation. Delta P over delta T. Um So this is approximately 3.5. The derivative Is exactly 3.5 when T equals two. Um But we're using this close in to approximate that derivative. And so delta P over delta T equals 3.5. So delta P Is equal to if we multiply both sides by Delta T. Delta P is equal to 3.5 times delta T. All right. Let's remember what T. And P. R. T. Is the number of years that have elapsed since 2000 and P is the percentage of power produced. So when T equals two, It's been two years since the year 2000. The rate at which the percentage of power being produced Changes with respect to a change in the time is 3.5. So when a little bit more time transpires beyond the two year mark, that would be your Delta T. So Delta T. A little bit more time, a portion of a year after the two year mark. So maybe another 20.1 years or another 0.2 years. Some small number Uh, change in the percentage of power produced is going to equal 3.5 times that change in the time. So as we move on past the two year mark, a small change in time Multiplied by 3.5 will give you approximately the change in the percentage of power produced. So the change in the percentage of power produced is going to be 3.5 times. Uh The increment change in the time.