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Application of Derivatives

Short Answer Type

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.

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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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Let r cm be the radius of the circle.
From given condition, rate of increase = 3 cm per second
$therefore space space space space space space dr over dt space equals space 3$
Let A be the area of circle, $therefore space space space space straight A space equals space πr squared$
$therefore space space space space space rate space of space increase space of space area space space equals space dA over dt space equals space straight d over dt left parenthesis πr squared right parenthesis space equals space 2 space πr space dr over dt space space space space space space space space$
$equals space 2 πr cross times 3 space equals space 6 πr$ $open square brackets because space space of space left parenthesis 1 right parenthesis close square brackets$
When r = 10, rate of increase of area = $6 straight pi cross times 10 space equals space 60 straight pi space cm squared divided by sec$

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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?