Question

If the percentage error in the measurement of surface area if solid sphere is 4 % then percentage error in the measurement of radius and volume

Answer 1

Answer:

% Error in Radius = 2%

% Error in Volume = 6%

Step-by-step explanation:

In the question,

The percentage error in the measurement of Surface Area of solid = 4%

So,

We know that,

Surface Area of Sphere is = 4πr²

Now,

$\frac{\delta S}{S}=2\left(\frac{\delta r}{r}\right)\\4=2\left(\frac{\delta r}{r}\right)\\2=\left(\frac{\delta r}{r}\right)$

Therefore, the percentage error in the measurement of radius is 2%.

Now,

Volume of Sphere is,

$V=\frac{4}{3}\pi r^{3}$

Therefore,

$\frac{\delta V}{V}\times 100=3\left(\frac{\delta r}{r}\times 100\right)\\\frac{\delta V}{V}\times 100=3(2\%)=6\%\\\frac{\delta V}{V}\times 100=6\%$

Therefore, the Percent Error in the measurement of Volume is 6%.

Answer 2

Given: If the percentage error in the measurement of surface area if solid sphere is 4 %.

To find: The percentage error in the measurement of radius and volume.

Solution: The percentage error in the measurement of radius and volume is 2% and 6%, respectively.

The surface area of a sphere can be given by the following formula.

$S = 4\pi r^{2}$

Here, S is the surface area and r is the radius of the sphere. Since the surface area is directly proportional to the radius of the sphere, the percentage error can be written as follows.

$\frac{\delta S}{S} = 2 \frac{\delta r}{r}$

$4 = 2 \frac{\delta r}{r}$

$2 = \frac{\delta r}{r}$

Thus, the percentage error in the measurement of radius is 2%. Now, the volume of a sphere can be found using the following formula.

$V = \frac{4}{3} \pi r^{3}$

Since the volume is directly proportional to the cube of the radius of the sphere, the percentage error can be written as follows.

$\frac{\delta V}{V} = 3 * \frac{\delta r}{r}$

     $= 3 * 2$

     $= 6$

Thus, the percentage error in the measurement of volume is 6%.

Therefore, the percentage error in the measurement of radius and volume is 2% and 6%, respectively.