Percentage error in the measurment of height and radius of cylinder are x and y respectively. Find percentage error in the measurement of volume. Which of the two measurements height or radius needs more attention? Why?
Answers
Given :-
↣ Percentage error in measurement of height ( h ) of cylinder is x
↣ Percentage error in the measurement of radius ( r ) of cylinder is y
To find :-
↣ percentage error in the measurement of volume of cylinder
↣ Which of the two measurements height or radius needs more attention and why ?
Knowledge required :-
↪ Volume of cylinder = π r² h
( where r is the radius and h is the height of cylinder )
↪ Error of a product or quotient
Let, Z = AB
then,
Δ Z / Z = ( Δ A / A ) + ( Δ B / B )
↪ Error in case of a measured quantity raised to a power
Generally if
Z = Aᵃ Bᵇ
then,
Δ Z / Z = a ( Δ A / A ) + b ( Δ B / B )
Solution :-
According to the given information
(Δ h / h) × 100 % = x .... eqn(2)
(Δ r / r) × 100 % = y .... eqn(1)
we have to find
(Δ V / V) × 100 %
→ As we know,
Volume of cylinder ( V ) = π r² h
so,
(ΔV / V) = 2 ( Δ r / r ) + ( Δ h / h )
( multiplying by 100 % both sides )
(ΔV / V) ×100%= 2(Δr/r)×100%+(Δh/h)×100%
( using eqn (1) and (2)
(ΔV / V) × 100 % = 2 ( y ) + x
so,
(ΔV / V)×100% = 2 y + x (Ans.)
●measurement of Radius needs more attention because its error will be doubled in this Case .