Question

(a) if a is the area of a circle with radius r and the circle expands as time passes, find \\(\frac{da}{dt}\\) in terms of \\(\frac{dr}{dt}).
\\(\frac{da}{dt}=( )\frac{dr}{dt})
(b) suppose oil spills from a ruptured tanker and spreads in a circular pattern. if the radius of the oil spill increases at a constant rate of 2 m/s, exactly how fast (in \\(m^{2}/s\\)) is the area of the spill increasing when the radius is 26 m?
( ) \\(m^{2}/s
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Explanation:

Step1: Recall area formula of circle

The area formula of a circle is $A = \pi r^{2}$.

Step2: Differentiate with respect to time $t$

Using the chain - rule, $\frac{dA}{dt}=\frac{dA}{dr}\cdot\frac{dr}{dt}$. Differentiating $A = \pi r^{2}$ with respect to $r$ gives $\frac{dA}{dr}=2\pi r$. So $\frac{dA}{dt}=2\pi r\frac{dr}{dt}$.

Step3: Solve part (b)

We know that $\frac{dr}{dt}=2$ m/s and $r = 26$ m. Substitute these values into the formula $\frac{dA}{dt}=2\pi r\frac{dr}{dt}$. Then $\frac{dA}{dt}=2\pi\times26\times2$.

Step4: Calculate the result

$\frac{dA}{dt}=104\pi$ m²/s.

Answer:

(a) $2\pi r$
(b) $104\pi$