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Tomtit [17]
3 years ago
5

PLZ HELP ME!!!!!

Mathematics
1 answer:
saveliy_v [14]3 years ago
3 0
5b-2=18
.............
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A circle is growing so that the radius is increasing at the rate of 3 cm/min. How fast is the area of the circle changing at the
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The area is growing at a rate of \frac{dA}{dt} =226.2 \,\frac{cm^2}{min}

Step-by-step explanation:

<em>Notice that this problem requires the use of implicit differentiation in related rates (some some calculus concepts to be understood), and not all middle school students cover such.</em>

We identify that the info given on the increasing rate of the circle's radius is 3 \frac{cm}{min} and we identify such as the following differential rate:

\frac{dr}{dt} = 3\,\frac{cm}{min}

Our unknown is the rate at which the area (A) of the circle is growing under these circumstances,that is, we need to find  \frac{dA}{dt}.

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We now apply the derivative operator with respect to time (\frac{d}{dt}) to this equation, and use chain rule as we find the quadratic form of the radius:

\frac{d}{dt} [A=\pi\,r^2]\\\frac{dA}{dt} =\pi\,*2*r*\frac{dr}{dt}

Now we replace the known values of the rate at which the radius is growing ( \frac{dr}{dt} = 3\,\frac{cm}{min}), and also the value of the radius (r = 12 cm) at which we need to find he specific rate of change for the area :

\frac{dA}{dt} =\pi\,*2*r*\frac{dr}{dt}\\\frac{dA}{dt} =\pi\,*2*(12\,cm)*(3\,\frac{cm}{min}) \\\frac{dA}{dt} =226.19467 \,\frac{cm^2}{min}\\

which we can round to one decimal place as:

\frac{dA}{dt} =226.2 \,\frac{cm^2}{min}

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