Question

The radius of a spherical balloon is measured as 20 inches, with a possible error of 0.01 inch. Use differentials to approximate the maximum possible error in calculating the following:

(a) the possible propagated error in computing the volume of the sphere.
(b) the possible propagated error in computing the surface area of the sphere.

Answer 1

a)
`\Delta V \approx 50.265\,in^(3)`, b)
`\Delta A_(s) \approx 5.027\,in^(2)`

Step-by-step explanation:

a) The volume of the sphere is:

`V = (4)/(3)\pi\cdot r^(3)`

The total differential of the volume of the sphere is:

`\Delta V = 4\pi\cdot r^(2)\,\Delta r`

`\Delta V = 4\pi \cdot (20\,in)^(2)\cdot (0.01\,in)`

`\Delta V \approx 50.265\,in^(3)`

b) The surface area of the sphere is:

`A_(s) = 4\pi\cdot r^(2)`

The total differential of the surface area of the sphere is:

`\Delta A_(s) = 8\pi \cdot r\,\Delta r`

`\Delta A_(s) = 8\pi \cdot (20\,in)\cdot (0.01\,in)`

`\Delta A_(s) \approx 5.027\,in^(2)`