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Application of Derivatives

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Short Answer Type

31.
The length x of a rectangle is decreasing at the rate of 3 cm/minute and the width y is increasing at the rate of 2 cm/minute. When x = 10 cm and y = 6 cm, find the rate of change of (a) the perimeter and (b) the area of rectangle

Since the length x is decreasing and the width y is increasing
$therefore space space space space space dx over dt space equals space minus 3 space space cm divided by minute space space space and space space dy over dt space equals space 2 cm divided by minute$ ...(1)
(a) The perimeter P of the rectangle is given by
$straight P space equals space 2 left parenthesis straight x plus straight y right parenthesis$
$therefore space space space space space space dP over dt space equals space 2 space open parentheses dx over dt plus dy over dt close parentheses space equals space 2 left parenthesis negative 3 plus 2 right parenthesis space equals space minus 2 space cm divided by minute$
(b) The area A of the rectangle is given by
A = xy
$therefore space space space space space space dx over dt space equals space straight x dy over dt plus straight y dx over dt space equals space 10 left parenthesis 2 right parenthesis space plus space 6 left parenthesis negative 3 right parenthesis space space space space space space space space space space space open square brackets because space space space straight x space equals space 10 comma space space straight y space equals space 6 space and space because space space of space left parenthesis 1 right parenthesis close square brackets$
$equals 20 minus 18 space equals space 2 space cm squared divided by minute$

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32.

The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minutc. When x = 8 cm and y = 6 cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle.

77 Views

Long Answer Type

33.

The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base?

88 Views

Short Answer Type

34.

A particle moves along the curve 6y = x³ + 2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.

141 Views

Long Answer Type

35.

At what points of the ellipse 16x² + 9y² = 400, does the ordinate decrease at the same rate at which the abscissa increases?

289 Views

36.

A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4m away from the wall?

85 Views

Short Answer Type

37.

Sand is pouring from a pipe at the rate of 12 cm³/s. The falling sand forms a cone on the ground in such a way what the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand-cone increasing when the height is 4 cm?

78 Views

Long Answer Type

38. A water tank has the shape of an inverted right circular cone with its axis vertical and vertex lowermost. Its semi-vertical angle is tan^–1 (0.5). Water is poured into it at a constant rate of 5 cubic metre per hour. Find the rate at which the level of the water is rising at the instant when the depth of water in the tank is 4 m.

139 Views

Short Answer Type

39.

A man 2 metres high walks at a uniform speed of 6 metre/sec away from a lamp-post 6 metres high. Find the rate at which the length of his shadow increases.

77 Views

40.

A man of height 2 metres walks at a uniform speed of 5 km/h away from a lamp post which is 6 metres high. Find the rate at which the length of his shadow increases.

170 Views