The radius of a sphere is 5.3± 0.1 cm. Calculate the percentage error in its volume.

To find the percentage error in the volume of the sphere, we start by looking at the relation between the volume and the radius of a sphere.

The volume $V$ of a sphere of radius $r$ is given by the formula:
$V=\frac{4}{3}\pi r^3$

When calculating errors, the constants (like $\frac{4}{3}$ and $\pi$) do not contribute to the fractional or percentage error. Therefore, the relative error in volume $\frac{\Delta V}{V}$ is related to the relative error in radius $\frac{\Delta r}{r}$ by the power rule of error propagation:
$\frac{\Delta V}{V}=3·\frac{\Delta r}{r}$

To find the percentage error, we multiply the relative error by 100:
$\text{Percentage error in}V=3·\left(\frac{\Delta r}{r}·100\right)$

From the given data, the radius of the sphere is:
$r\pm \Delta r=5.3\pm 0.1\text{cm}$
This gives us $r=5.3\text{cm}$ and the absolute error $\Delta r=0.1\text{cm}$.

Now, substitute these values into the percentage error formula:
$\text{Percentage error in}V=3·\left(\frac{0.1}{5.3}·100\right)$

Let's calculate the value inside the parentheses first:
$\frac{10}{5.3}\approx 1.8868\%$

Now multiply this result by 3:
$3·1.8868\%\approx 5.66\%$
Rounding this to one decimal place gives 5.7%.

Thus, the percentage error in the volume of the sphere is approximately 5.7%.