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The current $i$ (in $A$ ) in a certain electric circuit is a function of the time $t$ (in s) and a variable resistor $R$ (in $\Omega$ ), given by $i=\frac{6.0 \sin 0.01 t}{R+0.12} .$ Find $i$ for $t=0.75 \mathrm{s}$ and $R=1.50 \Omega$.

$0.028 \mathrm{A}$

Calculus 3

Chapter 29

Partial Derivatives and Double Integrals

Section 1

Functions of Two Variables

Partial Derivatives

Johns Hopkins University

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

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so in this question, were given that P is equal to our I squared under arrest ified dp dt or the derivative of this right here. We were also given some other conditions. So we're looking for that AT T is equal to 1/3. We're also told that I is equal to sign of four pi ti. So first, let's rewrite this equation right here plugging in, um, our expression for I So this is gonna be equivalent to P is equal to are, which is a constant times sign square since we have ice crowd right here, So sine squared of four high t Um and now we're ready to actually take the derivative. So D p d t, um, is gonna be equal to first. Our outer function is sign of four pi t squared. So let's use the chain rule. Let's multiply by two. And this attract one from the exponents. So that's gonna leave us with two r Sign of four pi team. Um, so next we have signed. So we're gonna take the druid of that, and that's gonna give us co sign co sign of four pi ti. Um and finally, our most inter function is for pi ti through. That is just gonna be four pi times for pie. So this is equal to yeah, hi are and again or is just a constant times sign of four pi ti times co sign of fork i t. So now we're actually ready to plug in what we're given in this question. So we know that we just write this on the side. That t is equal to 1/3 and our is equal to 1000. So it's part of that into our equation here. So we're gonna get that DP duty AT T is equal to 1/3 is equal to eat Pie times 1000 Um, a times sign of four pi over three times co sign of four pi over three. Um, if you multiply this, you get about 10,000 882 0.8 watts. So this right here is our final answer.

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