Question

The diameter of a sphere is measured as \[1.71cm\] using an instrument with a least count of \[0.01cm\]. The percentage error in surface area is

Answer 1

Hint:
Here, we will first find the actual value of the diameter by using the least count value. Then by using the formula of the surface area of a sphere we will find the surface area with the measured value of diameter. By using the percentage error formula, we will find the percentage error in surface area.

Formula Used:
We will use the following formula:
1) Actual value of a physical quantity is given by the formula \[d = {d^{'}} \pm L.C.\]
2) Percentage error is calculated by the formula \[{\text{Percentage Error}} = \dfrac{{{\text{Least Count}}}}{{{\text{Measured Value}}}} \times 100\% \] where \[d\] is the actual value of the diameter, \[{d^{'}}\] is the measured value of the diameter, \[L.C.\] is the least count.
3) Surface Area of a sphere is given by the formula \[S = 4\pi {r^2}\] where \[r\] is the radius of the sphere.

Complete step by step solution:
 We are given that the diameter of a sphere is measured as \[1.71cm\].
We are given that the least count when measuring the diameter of a sphere is \[0.01cm\].
Actual value of a physical quantity is given by adding or subtracting the least count and the measured value of a physical quantity. Therefore, we get
\[d = d' \pm L.C.\]
\[ \Rightarrow d = 1.71 \pm 0.01cm\]
Now we will calculate the surface area of the sphere.
We know that the radius of a sphere is half of the diameter of a sphere i.e., \[r = \dfrac{d}{2}\].
Substituting \[r = \dfrac{d}{2}\] in the formula \[S = 4\pi {r^2}\], we get
Surface Area of a sphere \[ = 4\pi {\left( {\dfrac{d}{2}} \right)^2}\]
Applying the exponent on the terms, we get
\[ \Rightarrow \] Surface Area of a sphere \[ = 4\pi \times \dfrac{{{d^2}}}{4}\]
By dividing the number 4, we get
\[ \Rightarrow \] Surface Area of a sphere\[ = \pi {d^2}\]
Since the surface area varies only when there is an error in the radius of a sphere. Therefore using the formula \[{\text{Percentage Error}} = \dfrac{{{\text{Least Count}}}}{{{\text{Measured Value}}}} \times 100\% \], we get
\[ \Rightarrow \] Percentage error in surface area \[ = \dfrac{{{\text{Least Count}}}}{{\left( {\dfrac{{d'}}{2}} \right)}} \times 100\% \]
\[ \Rightarrow \] Percentage error in surface area \[ = 2\left( {\dfrac{{{\text{Least Count}}}}{{{d^{'}}}}} \right) \times 100\% \]
Substituting the values of the measured value of the diameter, we get
\[ \Rightarrow \] Percentage error in surface area \[ = 2\left( {\dfrac{{0.01}}{{1.71}}} \right) \times 100\% \]
Simplifying the expression, we get
\[ \Rightarrow \] Percentage error in surface area \[ = 0.01169 \times 100\% \]
\[ \Rightarrow \] Percentage error in surface area \[ = 1.169\% \]

Therefore, the percentage error in surface area is \[1.169\% \].

Note:
Whenever we are measuring any physical quantity using an instrument, then there occurs an error in the measurements. We know that the least count is defined as the error made by the instrument while measuring the physical quantity. The least count is the minimum value in the measuring instrument. Percentage error is defined as the ratio of the least count to the measured value of the physical quantity multiplied by hundreds. It cannot be calculated for the actual value of the physical quantity.