## APPLICATION PROBLEMS OF RATE OF CHANGE IN CALCULUS

Application Problems of Rate of Change in Calculus :

In this section, we will see some practice questions on application problems of rate of change.

## Application Problems on Rate of Change - Example with Solution

Example 1 :

Newton's law of cooling is given by θ = θ₀° e⁻kt, where the excess of temperature at zero time is θ₀° C and at time t seconds is θ° C. Determine the rate of change of temperature after 40 s given that θ₀ = 16° C and k = -0.03.(e1.2  =  3.3201)

Solution :

Newton's law of cooling θ  =  θ₀° e⁻kt

θ₀°  =  16° C

k  =  -0.03

rate of change of temperature with respect to time

dθ/dt = - k θ₀° e^(⁻kt)

t  =  40

dθ/dt  =  - (-0.03) (16)e^(⁻0.03) (40)

=  0.48  e⁻1.2

=  0.48 (3.3201)

=  1.5936° C/s

Example 2 :

The altitude of a triangle is increasing at a rate of 1 cm/min while the area of the triangle is increasing at a rate of 2 cm²/min. At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 100 cm².

Solution :

Let "b" and "h" be the base and height of the triangle ABC

Area of triangle ABC

A  =  (1/2) b h

h  =  10      area  =  100

100  =  (1/2) x b x 10

b  =  (100 x 2)/10

b  =  20

dA/dt  =  (1/2) [b (dh/dt) + h (db/dt)]

h (db/dt)  =  2 (dA/dt) - b (dh/dt)

db/dt  =  (2/h) (dA/dt) - (b/h) (dh/dt)

db/dt  =  (2/10) (2) - (20/10) (1)

db/dt  =  (4/10) - (20/10)

db/dt  =  -16/10

db/dt  =  -1.6 cm/min

Example 3 :

At noon, ship A is 100 km west of ship B. Ship A is sailing east at 35 km/hr and ship B us sailing north at 25 km/hr. How fast is the distance between the ship changing at 4.00 p.m

Solution :

Let P and Q are the starting position of the ships A and B. Let x, y and z be the distance between QA and QB and AB respectively.

z2  =  x2 + y2

Here x, y and z are changing with respect to time.

2z (dz/dt)  =  2x (dx/dt) + 2y (dy/dt)

Dividing each side by 2.

z (dz/dt)  =  x (dx/dt) + y (dy/dt)

dx/dt  =  speed of ship A  =  35 km/hr

dy/dt  =  speed of B = 25 km/hr

x  =  40,  y  =  100

z  =  √(40)2 + (100)2

z  =  √1600 + 10000

z  =  √2600

z  =  20√29

20√29 (dz/dt)  =  40 (35) + (100) (25)

20√29 (dz/dt)  =  1400 + 2500

20√29 (dz/dt)  =  3900

dz/dt  =  3900 /(20√29)

dz/dt  =  195/√29

There the rate of change of distance between two ships is 195/√29 km/hr. We hope that the students would have understood how to solve word problems in rate of change.

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

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

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

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

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

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

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

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

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

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6 