Related Rate Worksheet

**Page 1**

1.

Find the value(s) of $x$ for which the rate of change of $y$ = $\frac{{x}^{3}}{3}+4{x}^{2}+16x$ with respect to $x$ is 9.

a. | - 2 | ||

b. | - 7, 1 | ||

c. | - 7, - 1 | ||

d. | 7, - 1 |

The rate of change of

[Substitute

(

[Factorize.]

[Solve.]

Correct answer : (3)

2.

A small stone is dropped into a quiet pond and circular ripples spread over the surface of water. The radius of each of these ripples increase at the rate of 28 inches per second. Find the rate at which the area inside the circle is increasing at the instant the radius is 7 ft.

a. | $\frac{98}{3}$$\pi $ ft²/s | ||

b. | 392$\pi $ ft²/s | ||

c. | $\frac{49}{3}$$\pi $ ft²/s | ||

d. | $\frac{14}{3}$$\pi $ ft²/s |

Area of the water ring of radius

The rate of increase of radius

=

[One inch =

The rate of increase in area A is

[Definition.]

=

[Substitute A =

= 2

At

= 2

[Substitute

=

[Simplify.]

Correct answer : (1)

3.

The side length of a square increases at the rate of 4 cm/s. At what rate is the area of the square increasing when its side length is 40 cm?

a. | 8 cm²/s | ||

b. | 160 cm²/s | ||

c. | 320 cm²/s | ||

d. | 4 cm²/s |

The area of square of side

The rate of increase of side

The rate of increase of area A =

[Definition.]

=

[Substitute A =

= 2

At

= 2(40)(4)

[Substitute

= 320 cm²/sec

Correct answer : (3)

4.

The volume of a metallic hollow sphere is constant. Its outer radius is increasing at the rate of 2 cm/s. Find the rate at which its inner radius is increasing if the outer and inner radius of the sphere are 16 cm and 8 cm respectively.

a. | 8 cm/s | ||

b. | 6 cm/s | ||

c. | 4 cm/s | ||

d. | 10 cm/s |

[Formula.]

[As V is constant.]

[Substitute V from step1.]

[Simplify.]

[Simplify.]

At R = 16 cm,

[Substitute

=

Correct answer : (1)

5.

A gas balloon contains 1800 cubic ft. of gas at a pressure of 90 lb per ft². The pressure is decreasing at the rate of 0.1 lb per ft². per second. Find the rate at which the volume increases, if the gas obeys Boyle's law, PV = constant.

a. | 20 ft³/s | ||

b. | 2 ft³/s | ||

c. | 18 ft³/s | ||

d. | 180 ft³/s |

The rate of change of P is

PV = constant

[BoyleÃ¢â‚¬â„¢s law.]

[Take derivative.]

P(

[Product rule of derivative.]

At P = 90 lb per ft², V = 1800 ft³

So

[Substitute the values of P, V,

= 2 ft³/s

[Simplify.]

So , the rate of increase of the volume of the balloon =

Correct answer : (2)

6.

An airplane at an altitude of 900 m flying horizontally with a speed of 250 m/sec passes directly over an observer. Find the rate at which the plane approaches the observer, when it is at 936 m away from the observer.

a. | $\frac{1285}{18}$ m/s | ||

b. | $\frac{\sqrt{66096}}{900}$ m/s | ||

c. | $\frac{32125}{468}$ m/s | ||

d. | $\frac{\sqrt{66096}}{90}$ m/s |

Here A is the position of the observer.

[Figure.]

B is the position of the airplane at some instant.

[Figure.]

AC is the horizon.

[Figure.]

BC =

Let AB =

The speed of the airplane =

[From the right triangle ABC.]

936

[When

[Simplify.]

[Solve for

[From step 7.]

2

At

=

=

[Simplify.]

Correct answer : (3)

7.

If the semi vertical angle of a right circular cone is 45^{o} and the rate of change of volume of the cone is $\pi $${r}^{k}\frac{dr}{dt}$, then find the value of ($k$ + 2)($k$ + 5)($k$ + 4).

a. | 40 | ||

b. | 168 | ||

c. | 160 | ||

d. | 80 |

The base radius

[From the right triangle ABC.]

Volume of the cone = V =

V =

[Substitute

The rate of change of volume =

[Definition.]

=

[Substitute from step 4.]

=

[Solve for

Therefore the value of

Correct answer : (2)

8.

$x$ = $\sqrt{4}\mathrm{cos}5\theta $ and $y$ = $\sqrt{4}\mathrm{sin}5\theta $, where 0 ≤ $\theta $ ≤ $\pi $. Find the value of $\theta $ at which the rate of change of $x$ and $y$ with respect to $\theta $ are equal.

a. | $\frac{1}{20}$ | ||

b. | 3$\pi $ | ||

c. | $\frac{\pi}{20}$ | ||

d. | $\frac{3\pi}{20}$ |

[Substitute

[Substitute

If

tan 5

[Solve tan

[Simplify.]

Correct answer : (4)

9.

The rate of change of the circumference C of a circle with respect to its area A is $k$ $\sqrt{\frac{\pi}{\mathrm{A}}}$. Find the value of 9$k$² + 8$k$.

a. | $\frac{41}{16}$ | ||

b. | 52 | ||

c. | 17 | ||

d. | 44 |

Area of circle with radius

Circumference of circle with radius

C

[Square on both sides.]

C

[Substitute

2C

[Substitute

=

[Substitute

=

9

[Substitute the value of

Correct answer : (3)

10.

The side length of an equilateral triangle is $l$ cm. If the rate of change of area of the incircle with respect to $l$ is $\frac{k\pi l}{6}$, then find the value of 9k + 18.

a. | 27 | ||

b. | 18 | ||

c. | 9 | ||

d. | 28 |

Side of the equilateral triangle =

The radius of incircle of the triangle =

[Formula.]

The area of the incircle = A = πr

=

[Substitute the value of

=

[Simplify.]

The rate of change of the area of the incircle with respect to the side

Therefore

9

Correct answer : (1)