Question

The radius of a spherical balloon is measured as 20 inches, with a possible error of 0.03 inch. Use differentials to approximate the maximum possible error in calculating the following. a) Calculate the volume of the sphere. b) Calculate the surface area of the sphere. c) Calculate the relative errors in part a) and b).

Answer 1

Answer:

a) $V = 33510.322\,in^{3}$, b) $A_{s} = 5026.548\,in^{2}$, c) $\% V = 0.450\,\%$, $\%A_{s} = 0.300\,\%$.

Step-by-step explanation:

The volume and the surface area of the sphere are, respectively:

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

$A_{s} = 4\pi \cdot r^{2}$

a) The volume of the sphere is:

$V = \frac{4}{3}\pi \cdot (20\,in)^{3}$

$V = 33510.322\,in^{3}$

b) The surface area of the sphere is:

$A_{s} = 4\pi \cdot (20\,in)^{2}$

$A_{s} = 5026.548\,in^{2}$

c) The total differentials for volume and surface area of the sphere are, respectively:

$\Delta V = 4\pi\cdot r^{2}\,\Delta r$

$\Delta V = 4\pi \cdot (20\,in)^{2}\cdot (0.03\,in)$

$\Delta V = 150.796\,in^{3}$

$\Delta A_{s} = 8\pi\cdot r \,\Delta r$

$\Delta A_{s} = 8\pi \cdot (20\,in)\cdot (0.03\,in)$

$\Delta A_{s} = 15.080\,in^{2}$

Relative errors are presented hereafter:

$\%V = \frac{\Delta V}{V}\times 100\%$

$\%V = \frac{150.796 \,in^{3}}{33510.322\,in^{3}}\times 100\,\%$

$\% V = 0.450\,\%$

$\% A_{s} = \frac{\Delta A_{s}}{A_{s}}\times 100\,\%$

$\% A_{s} = \frac{15.080\,in^{2}}{5026.548\,in^{2}}\times 100\,\%$

$\%A_{s} = 0.300\,\%$