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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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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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Let r be the radius of the balloon.
$therefore space space space space space space dr over dt space equals space 10 space cm divided by sec$ . ...(1)
Let S be the surface area of the balloon.
$therefore space space space space space space space space space straight S space equals space 4 space straight pi space straight r squared therefore space space space space space space space space ds over dt space equals space straight d over dt left parenthesis 4 πr squared right parenthesis space equals space 4 space straight pi space straight d over dt left parenthesis straight r squared right parenthesis space equals space 4 space straight pi space open parentheses 2 straight r space dr over dt close parentheses equals space 8 space straight pi space straight r space dr over dt space space space space space space space space space space space space space space space space space space space space space space$
$equals space 8 πr space cross times space 10 space equals space 80 space straight pi space straight r space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space space of space left parenthesis 1 right parenthesis close square brackets$
When r = 15, $dS over dt space equals space 80 space cross times space straight pi space cross times space 15 space equals space 1200 space straight pi$
$therefore space space space space space space rate space of space increase space of space surface space area space equals space 1200 space straight pi space sq. space cm divided by sec.$

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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?