Question - 1 : - The side of a square sheet is increasing at the rate of 4 cm per minute. At what rate is the area increasing when the side is 8 cm long?

Answer - 1 : -

Given the side of a square sheet is increasing at the rate of 4cm per minute.

To find rate of area increasing when the side is 8 cm long

Let the side of the given square sheet be x cm at any instanttime.

Then according to the given question, we can write as

Question - 2 : - An edge of a variable cube is increasing at the rate of 3 cm per second. How fast is the volume of the cube increasing when the edge is 1 cm long?

Answer - 2 : -

Given the edge of a variable cube is increasing at the rate of 3 cm per second.

To find rate of volume of the cube increasing when the edge is 1 cm long

Let the edge of the given cube be x cm at any instant time.

Then according to the given question we can write as

Question - 3 : - The side of a square is increasing at the rate of 0.2 cm/sec. Find the rate of increase of the perimeter of the square.

Answer - 3 : -

Given the side of a square is increasing at the rate of 0.2 cm/sec.

To find rate of increase of the perimeter of the square

Let the edge of the given cube be x cm at any instant time.

Then according to the given question, we can write as

Question - 4 : - The radius of a circle is increasing at the rate of 0.7 cm/sec. What is the rate of increase of its circumference?

Answer - 4 : -

Given the radius of a circle is increasing at the rate of 0.7 cm/sec.

To find rate of increase of its circumference

Let the radius of the given circle be r cm at any instant time.

Then according to the given question, we can write as

Hence the rate of increase of the circle’s circumference will be1.4 π cm/sec

Question - 5 : - The radius of a spherical soap bubble is increasing at the rate of 0.2 cm/sec. Find the rate of increase of its surface area, when the radius is 7 cm

Answer - 5 : -

Given the radius of a spherical soap bubble is increasing at the rate of 0.2 cm/sec.

To find rate of increase of its surface area, when the radius is 7 cm

Let the radius of the given spherical soap bubble be r cm at any instant time.

Then according to the given question we can write as

Rate of radius of the spherical soap bubble is increasing

Hence the rate of increase of its surface area, when the radiusis 7 cm is 11.2 π cm²/sec

Question - 6 : - A balloon which always remains spherical is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon is increasing when the radius is 15 cm.

Answer - 6 : -

Given spherical balloon inflated by pumping in 900 cubic centimetres of gas per second

To find the rate at which the radius of the balloon is increasing when the radius is 15 cm

Let the radius of the given spherical balloon be r cm and let V be the volume of the spherical balloon at any instant time

Then according to the given question,

As the balloon is inflated by pumping 900 cubic centimetres of gas per second hence the rate of volume of the spherical balloon increases by

Question - 7 : - The radius of an air bubble is increasing at the rate of 0.5 cm/sec. At what rate is the volume of the bubble increasing when the radius is 1 cm?

Answer - 7 : -

Given radius of an air bubble is increasing at the rate of 0.5cm/sec

To find the rate at which the volume of the bubble increasingwhen the radius is 1 cm

Let the radius of the given air bubble be r cm and let V be thevolume of the air bubble at any instant time

Then according to the given question,

Question - 8 : - A man 2 metres high walks at a uniform speed of 5 km/hr. away from a lamp – post 6 metres high. Find the rate at which the length of his shadow increases.

Answer - 8 : -

Question - 9 : - A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm/sec. At the instant when the radius of the circular wave is 10 cm, how fast is the enclosed area increasing?

Answer - 9 : -

Given a stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm/sec.

To find the instant when the radius of the circular wave is 10 cm, how fast is the enclosed area increasing

Let r be the radius of the circle and A be the area of the circle

When stone is dropped into the lake waves moves in circle at speed of 4cm/sec. i.e., radius of the circle increases at a rate of 4cm/sec

Question - 10 : - man 160 cm tall walks away from a source of light situated at the top of a pole 6 m high, at the rate of 1.1m/sec. How fast is the length of his shadow increasing when he is 1m away from the pole?

Answer - 10 : -

Given a man 160cm tall walks away from a source of lightsituated at the top of a pole 6 m high, at the rate of 1.1m/sec

To find the rate at which the length of his shadow increaseswhen he is 1m away from the pole

Let AB be the lamp post and let MN be the man of height 160cm or1.6m.

Let AL = l meter and MS be the shadow of the man

Let length of the shadow MS = s (as shown in the below figure)

Hence the rate at which the length of his shadow increases by0.4 m/sec, and it is independent to the current distance of the man from thebase of the light.

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RD Chapter 13 Derivative as a Rate Measurer Ex 13.2 Solutions

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