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11.
A ballon which always remain spherical, has a variable diameter $3 over 2 left parenthesis 2 straight x plus 3 right parenthesis$ . Determine the rate of charge of its volume with respect to x.

Let r be the radius of the balloon.
From given condition, diameter = $3 over 2 left parenthesis 2 straight x plus 3 right parenthesis$
$therefore space space space space space space 2 straight r space equals space 3 over 2 left parenthesis 2 straight x plus 3 right parenthesis space space space space space space space space space space space space rightwards double arrow space space space space straight r space equals space 3 over 4 left parenthesis 2 straight x plus 3 right parenthesis$
Let V be the volume of spherical balloon.
$therefore space space space space space straight V space equals space fraction numerator 4 straight pi over denominator 3 end fraction straight r cubed space equals space fraction numerator 4 straight pi over denominator 3 end fraction. space space space 27 over 64 left parenthesis 2 straight x plus 3 right parenthesis cubed space space space space space space space space space space space space space space rightwards double arrow space space space space straight V space equals space fraction numerator 9 straight pi over denominator 16 end fraction left parenthesis 2 straight x plus 3 right parenthesis cubed therefore space space space space rate space of space change space of space volume space straight w. straight r. straight t. space straight x space equals space dV over dx space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space straight d over dx open square brackets fraction numerator 9 straight pi over denominator 16 end fraction left parenthesis 2 straight x plus 3 right parenthesis cubed close square brackets space equals space fraction numerator 9 straight pi over denominator 16 end fraction. straight d over dx left parenthesis 2 straight x plus 3 right parenthesis cubed space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 9 straight pi over denominator 16 end fraction.3 space space left parenthesis 2 straight x plus 3 right parenthesis squared space. space 2 space equals space fraction numerator 27 space straight pi over denominator 8 end fraction left parenthesis 2 straight x plus 3 right parenthesis squared.$

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12.

The radius of a circle is increasing at 0.7 cm/sec. What is the rate of increase of its circumference when r = 4.9 cm?

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13.

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

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14.

The radius of a circle is increasing uniformly at the rate of 3 cm per second. Find the rate at which the area of the circle is increasing when the radius is 10 cm.

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15. A car starts from a point P at time t = 0 seconds and stops at point Q. The distance x, in metres, covered by it, in t seconds is given by

straight x space equals straight t squared open parentheses 2 minus straight t over 3 close parentheses.

Find the time taken by it to reach Q and also find distance between P and Q.

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16.

A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm per second. At the instant when the radius of the circular waves is 10 cm, how fast is the enclosed area increasing?

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17. A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 Cm, how fast is the enclosed area increasing?

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18.

The radius of a balloon is increasing at the rate of 10 cm per second. At what rate is the surface area of the balloon increasing when its radius is 15 cm?

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19.

The radius of an air-bubble is increasing at the rate of $1 half cm space per space second.$ At what rate is the volume of the bubble increasing when the radius is 1 cm?