Question

If the percentage error in the radius of a circle is 3, then the percentage error in its area is

• A. 6
• B. 3/2
• C. 2
• D. 4

Hint:

For any formula of the form $Z = kX^aY^bZ^c$, the maximum percentage error in $Z$ is given by the sum of the percentage errors in its components, each multiplied by the magnitude of its exponent: $%Z = |a|(%X) + |b|(%Y) + |c|(%Z)$.

Answer 1

The Correct Option is A

Step 1: Write the formula for the area of a circle.
The area $A$ of a circle with radius $r$ is given by the formula: \[ A = \pi r^2. \]

Step 2: Use the formula for relative error.
For a quantity $Z = kX^n$, the relative error is given by $\frac{\Delta Z}{Z} = n \frac{\Delta X}{X}$. In our case, $A = \pi r^2$, so the relative error in the area is: \[ \frac{\Delta A}{A} = 2 \frac{\Delta r}{r}. \]

Step 3: Convert to percentage error.
To find the percentage error, we multiply the relative error by 100. \[ \left(\frac{\Delta A}{A} \times 100%\right) = 2 \left(\frac{\Delta r}{r} \times 100%\right). \]

Step 4: Substitute the given percentage error in the radius.
We are given that the percentage error in the radius is 3%. \[ \text{Percentage error in Area} = 2 \times (3%). \] \[ \boxed{\text{Percentage error in Area} = 6%}. \]