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If a circular plate is heated uniformly, its area expands \(3 c\) times as fast as its radius, then the value of \(c\) when the radius is 6 units, is
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The correct answer is:
\(4 \pi\)
Let \(A\) sq. units in the area measure when the radius is \(r\) units.
their \(A=\pi r^2\)
Differentiate both side w.r.t ' \(t\) '
\(\frac{d A}{d t}=2 \pi r \frac{d r}{d t}\) ...(i))
We have, \(\frac{d A}{d t}=3 c \frac{d r}{d t}\)
From eqn (i), we get
\(3 c \cdot \frac{d r}{d t}=2 \pi r \cdot \frac{d r}{d t} \Rightarrow 3 c=2 \pi r\)
Now, \(c=\frac{2}{3} \pi(6)=4 \pi\) when \(r=6\)
their \(A=\pi r^2\)
Differentiate both side w.r.t ' \(t\) '
\(\frac{d A}{d t}=2 \pi r \frac{d r}{d t}\) ...(i))
We have, \(\frac{d A}{d t}=3 c \frac{d r}{d t}\)
From eqn (i), we get
\(3 c \cdot \frac{d r}{d t}=2 \pi r \cdot \frac{d r}{d t} \Rightarrow 3 c=2 \pi r\)
Now, \(c=\frac{2}{3} \pi(6)=4 \pi\) when \(r=6\)
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