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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

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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.

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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?

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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.

Here, 6y = x³ + 2. ...(1)
Differentiating both sides, w.r.t 't'
$6 dy over dt space equals space 3 straight x squared dx over dt$

But $dy over dt space equals space 8 dx over dt$ (given)

$therefore space space space space space space space 6 open parentheses 8 dx over dt close parentheses space equals space 3 straight x squared dx over dt space space space space space rightwards double arrow space space space space 16 space equals space straight x squared space space space space space space space space space space rightwards double arrow space space space space straight x space equals space 4 comma space space space minus 4$

$When space space space space straight x space equals space 4 space from space left parenthesis 1 right parenthesis comma space we space get comma space space space space space straight y space equals space fraction numerator left parenthesis 4 right parenthesis cubed plus 2 over denominator 6 end fraction space equals space 66 over 6 space equals space 11 When space space space straight x space equals space minus 4 space space from space left parenthesis 1 right parenthesis comma space we space get comma space space space straight y space equals space fraction numerator left parenthesis negative 4 right parenthesis cubed plus 2 over denominator 6 end fraction space equals space fraction numerator negative 62 over denominator 6 end fraction space equals space minus 31 over 3 therefore space space space space required space points space on space the space curve space are space left parenthesis 4 comma space 11 right parenthesis comma space space open parentheses negative 4 comma space space space space space space minus 31 over 3 close parentheses.$

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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?

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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?

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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?

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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.

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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.

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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.

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