Practicing Success
The percentage error in calculating the volume of a cubical box if an error of 1% is made in measuring the length of edges of the cube, is |
1% 2% 3% 3/2% |
3% |
Let x be the length of an edge of the cube and y be its volume. Then, $y=x^3$. Let $\Delta x$ be the error in x and $\Delta y$ be the corresponding error in y. $\frac{\Delta x}{x} \times 100=1$ [Given] $\Rightarrow \frac{d x}{x} \times 100=1$ [∵ dx ≅ Δx] ....(i) We have to find $\frac{\Delta y}{y} \times 100$. Now, $y=x^3 \Rightarrow \frac{d y}{d x}=3 x^2$ ∴ $d y=\frac{d y}{d x} d x$ $\Rightarrow d y=3 x^2 . d x \Rightarrow \frac{d y}{y}=\frac{3 x^2}{y} d x \Rightarrow \frac{d y}{y}=\frac{3 x^2}{x^3} d x ~~~\left[∵ y=x^3\right]$ $\Rightarrow \frac{d y}{y}=3 \frac{d x}{x} \Rightarrow \frac{d y}{y} \times 100=3\left(\frac{d x}{x} \times 100\right)=3$ [Using (i)] $\Rightarrow \frac{\Delta y}{y} \times 100=3$ $[∵ d y \cong \Delta y]$ So, there is 3% error in calculating the volume of the cube. |