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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?

Let ABC be an isosceles triangle in which AB = AC = x (say), BC = b.
From A, draw

AL perpendicular BC comma

so that

BL space equals space straight b over 2

In right

angle straight d space space space increment space ALB comma

AL space equals space square root of AB squared minus BL squared end root space equals space square root of straight x squared minus straight b squared over 4 end root

Let

increment

be area of

increment ABC.

therefore space space space space space space space increment space equals space 1 half BC space cross times space space AL space equals space 1 half straight b square root of straight x squared minus straight b squared over 4 end root

therefore space space space space fraction numerator straight d increment over denominator dt end fraction space equals space 1 half straight b. space fraction numerator 2 straight x over denominator 2 square root of straight x squared minus begin display style straight b squared over 4 end style end root end fraction dx over dt space equals space fraction numerator bx over denominator 2 square root of straight x squared minus begin display style straight b squared over 4 end style end root end fraction. space left parenthesis negative 3 right parenthesis space space space space space space space space space space space open square brackets because space space space dx over dt space equals space 3 space left parenthesis given right parenthesis close square brackets

equals space fraction numerator negative 3 space bx over denominator 2 square root of straight x squared minus begin display style straight b squared over 4 end style end root end fraction

space When space space space space space straight x space equals space straight b comma space space space fraction numerator straight d increment over denominator dt end fraction space equals space fraction numerator negative 3 straight b space cross times space straight b over denominator 2 square root of straight b squared minus begin display style straight b squared over 4 end style end root end fraction space equals space fraction numerator negative 3 straight b squared over denominator square root of 3 space straight b squared end root end fraction space equals space minus square root of 3 space straight b squared end root space equals space minus square root of 3 space straight b therefore space space space space required space rate space of space decrease space space equals space square root of 3 space straight b space cm squared divided by sec.

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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.

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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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