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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 :

xero099

Answer:

a) [tex]\Delta V \approx 50.265\,in^{3}[/tex], b) [tex]\Delta A_{s} \approx 5.027\,in^{2}[/tex]

Explanation:

a) The volume of the sphere is:

[tex]V = \frac{4}{3}\pi\cdot r^{3}[/tex]

The total differential of the volume of the sphere is:

[tex]\Delta V = 4\pi\cdot r^{2}\,\Delta r[/tex]

[tex]\Delta V = 4\pi \cdot (20\,in)^{2}\cdot (0.01\,in)[/tex]

[tex]\Delta V \approx 50.265\,in^{3}[/tex]

b) The surface area of the sphere is:

[tex]A_{s} = 4\pi\cdot r^{2}[/tex]

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

[tex]\Delta A_{s} = 8\pi \cdot r\,\Delta r[/tex]

[tex]\Delta A_{s} = 8\pi \cdot (20\,in)\cdot (0.01\,in)[/tex]

[tex]\Delta A_{s} \approx 5.027\,in^{2}[/tex]

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