Courses
Courses for Kids
Free study material
Offline Centres
More
Store Icon
Store
seo-qna
SearchIcon
banner

A physical quantity Z as a function of time is given as Z(t)=A32ekt , where k=0.1s1 . The measurement of A is 2.00% . If the error in the measurement of time is 1.25% , then the percentage error in the value of Z(t) at t=10s would be
(1) 4.25%
(2) 2.50%
(3) 3.75%
(4) 3.50%

Answer
VerifiedVerified
478.8k+ views
like imagedislike image
Hint : On performing any mathematical operation error always increases hence whether the quantity is added, subtracted, multiplied, divided the error in the quantities always gets added. Percentage error is the difference between the estimated value and the actual value in comparison to the actual value and is expressed as a percentage.

Complete step by step answer
From question,
 Z(t)=A32ekt
Taking logarithm to the base e on both sides,
 ln(Z(t))=32lnAkt
Differentiating both sides,
 d(ln(Z(t))) =d(32lnAkt)
Percentage error is the difference between the calculated value and the actual value with respect to the actual value and is expressed in a percentage format. In calculus dx represents the infinitesimal difference between the calculated value and the actual value.
 d(Z(t))Z(t)=32×dAAkdt
We know that performing any mathematical operation error always increases hence whether the quantity is added, subtracted, multiplied, divided the error in the quantities always gets added.
Hence,
 d(Z(t))Z(t)×100=32×dAA×100+kdttt
This is the percentage error in Z is the sum of percentage error in A multiplied by 32 and percentage error in t multiplied by kt .
 d(Z(t))Z(t)×100=32×2×100+0.1×1.25×10
As in the question, it is given that the percentage error in A is 2.00% , the error in the measurement of time is 1.25% , and as we have to find percentage error in the value of Z(t) when 10s has elapsed so t=10s .
 d(Z(t))Z(t)×100=3+1.25
 d(Z(t))Z(t)×100=4.25
Hence the percentage error in calculating Z is 4.25% .
Therefore, the correct answer to the given question is (A) 4.25% .

Note
For a mathematical equation, Z=axbycz the relative error in calculating Z will be xdaa+ydbb+zdcc but in this question we also proved how this equation came for your conceptual understanding. In physics calculus is an indispensable part so a student must know basic calculus to solve physics numerically.