Applications of derivatives, Mathematics

Assignment Help:

Applications of derivatives : At last, let's not forget about our applications of derivatives.

Example   Assume that the amount of air in a balloon at any time t is specified by

2245_application of derivation.png

Find out if the balloon is being filled with air or being drained of air at t =8 .

Solution

If the balloon is filled through air then the volume is increasing & if it's being drained of air then the volume will be decreasing.  In other terms, we have to get the derivative so that we can find out the rate of change of the volume at t = 8 .

It will require the quotient rule.

V ′(t ) =  2t (-2/3 ) ( 4t + 1) - 6t (1/3) ( 4) /(4t + 1)2

             = -16t (1/3)  + 2t  (-2/3) / (4t + 1)2

             =  -16t (1/3)  + 2/(t(2/3)) / (4t + 1)2

Note as well that we simplified the numerator more than usual here. It was only done to make the derivative simpler to evaluate.

The rate of change of the volume at t = 8 is then,

V ′ (8) =  -16 ( 2) + (2/4) / (33)2       (8) (1/3)  = 2       (8)(2/3)  = ( (8)(1/3 ) )2 =(2)2 =4

= -  63 /2178= -  7 /242

Thus, the rate of change of the volume at t = 8 is negative and hence the volume has to be decreasing. So air is being drained out of the balloon at t = 8 .


Related Discussions:- Applications of derivatives

Find the probability of having 53 sundays in leap year , Find the probabili...

Find the probability of having 53 Sundays in (i) a leap year                           (ii) a non leap year       (Ans:2/7 , 1/7 ) Ans:          An ordinary year has 365 da

Shares and dividends, at what price 6.25% rs 100 share be quoted when the m...

at what price 6.25% rs 100 share be quoted when the money is worth 5%

Determine the domain of the function, Determine or find out the domain of t...

Determine or find out the domain of the subsequent function. r → (t) = {cos t, ln (4- t) , √(t+1)} Solution The first component is described for all t's. The second com

Integrated marketing, #question.Explain its nature and how it influences th...

#question.Explain its nature and how it influences the integrated marketing communications mix and distinguish between tactical and strategic use of integrated marketing communicat

Which of the subsequent terms does not describe the number 9, Which of the ...

Which of the subsequent terms does NOT describe the number 9? Nine is NOT prime since it has 3 factors; 1, 3, and 9. Prime numbers have only 2 factors.

Example of uniform distribution, Q. Samantha wrote a computer program to r...

Q. Samantha wrote a computer program to randomly generate two-digit numbers between 00 and 99. Let X be the random 2 digit number generated by the computer. Find the distributio

Divide by 1-digit numbers, which experession can be used to check the quoti...

which experession can be used to check the quotient 646 divided by 3

Project, report on shares and dovidend using newspaper

report on shares and dovidend using newspaper

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd