**Rate of Change Word Problems in Calculus : **

In this section, let us look into some word problems using the concept rate of change.

The derivative can also be used to determine the rate of change of one variable with respect to another. A few examples are population growth rates, production rates, water flow rates, velocity, and acceleration.

A common use of rate of change is to describe the motion of an object moving in a straight line. In such problems, it is customary to use either a horizontal or a vertical line with a designated origin to represent the line of motion.

On such lines, movements in the forward direction considered to be in the positive direction and movements in the backward direction is considered to be in the negative direction.

**Example 1 :**

The radius of a circular plate is increasing in
length at 0.01 cm per second. What is the rate at which the area is increasing
when the radius is 13 cm?

**Solution :**

Let A and r be the area and the radius.

Rate of change = 0.01 cm

Area of circular plate A = Π r²

Differentiate with respect to r

dA/dt = Π 2r (dr/dt)

here r = 13 cm and dr/dt = 0.01 cm

dA/dt = Π 2r (dr/dt)

dA/dt = Π 2(13) (0.01)

dA/dt = Π (26) (0.01)

dA/dt = Π (0.26)

dA/dt = 0.26 Π cm²/sec

**Example 2 :**

A square plate is expanding uniformly each side is increasing at the constant rate of 1.5 cm/min. Find the rate at which the area is increasing when the side is 9 cm.

**Solution :**

Let "a" be the side of the square and "A" be the area of the square.

Here the side length is increasing with respect to time.

da/dt = 1.5 cm/min

Now we need to find the rate at which the area is increasing when the side is 9 cm.

That is,

We need to determine dA/dt when a = 9 cm.

Area of square = a^{2}

A = a^{2}

differentiate with respect to t

dA/dt = 2a (da/dt)

dA/dt = 2 (9) (1.5)

dA/dt = 18 (1.5)

dA/dt = 27 cm^{2}/min

**Example 3 :**

A stone thrown into still water causes a series of concentric ripples. If the radius of outer ripple is increasing at the rate of 5 cm/sec,how fast is the area of the distributed water increasing when the outer most ripple has the radius of 12 cm/sec.

**Solution :**

Let "A" be the area and "r" be the radius

Here the radius is increasing with respect to time.

Now we need to find dA/dt when radius = 12 cm/sec

dr/dt = 5 cm/sec

Area of circle = Π r²

A = Π r²

dA/dt = 2 Π r (dr/dt)

dA/dt = 2 Π (12) (5)

dA/dt = 24 Π (5)

dA/dt = 120 Π cm^{2}/sec

**Example 4 :**

The radius of a spherical balloon is increasing at the rate of 4 cm/sec. Find the rate of increases of the volume and surface area when the radius is 10 cm.

**Solution :**

Let V be the volume of spherical balloon and S be the surface area.

Here, we need to find dV/dt and dS/dt

dr/dt = 4 cm/sec and r = 10 cm

Volume of the spherical balloon (V) = (4/3) Π r^{3}

Differentiate with respect to t

dV/dt = (4/3) Π 3 r² (dr/dt)

dV/dt = (4/3) Π 3 (10)² (4)

dV/dt = 16 Π (10)²

dV/dt = 16 Π (100)

dV/dt = 1600 Π cm³/sec

Surface area of the spherical balloon S = 4 Π r²

differentiate with respect to t

dS/dt = 4 Π 2r (dr/dt)

dS/dt = 8 Π r (dr/dt)

dS/dt = 8 Π (10) (4)

dS/dt = 80 Π (4)

dS/dt = 320 Π cm^{2}/sec

**Example 5 :**

A balloon which remains spherical is being inflated be pumping in 90 cm³/sec. Find the rate at which the surface area of the balloon is increasing when the radius is 20 cm.

**Solution :**

Let "V" be the area and "r" be the radius of the balloon.

Now we need to find the surface area of the balloon when the radius is 20 cm

dV/dt = 90 cm³/sec.

Surface area of the balloon (S) = 4Π r^{2}

V = (4/3) Π r^{3}

dV/dt = (4/3) Π 3 r^{2} (dr/dt)

90 = (4/3) Π 3 (20)^{2} (dr/dt)

90 = 4 Π (400) (dr/dt)

90 = 1600 Π (dr/dt)

90/1600 Π = (dr/dt)

dr/dt = 9/160Π

Surface area of the spherical balloon = (S) = 4Π r^{2}

dS/dt = 4 Π (2r) dr/dt

dS/dt = 4 Π 2(20) (9/160Π)

dS/dt = 9 cm^{2}/sec

We hope that the students would have understood how to solve application problems on rate of change.

If you need any other stuff, please use our google custom search here.

HTML Comment Box is loading comments...

You can also visit our following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**