Question

The radius of a sphere is R = (22.2 ± 0.1)cm. The radius of the base of a cylinder is r = (12.0 ± 1.2)cm, and its height is h = (24.4 ± 1.1)cm. What is the total volume occupied by the sphere and by the cylinder (including the error)?

Answer 1

Given:

Radius of a sphere, R = (22.2 ± 0.1)cm

Radius of the base of a cylinder, r = (12.0 ± 1.2)cm

The height of the cylinder, h = (24.4 ± 1.1)cm

To find:

What is the total volume occupied by the sphere and by the cylinder (including the error)?

Solution:

Finding the volume (including error) occupied by the sphere:

We have,

R = 22.2 cm

ΔR = 0.1 cm

∴ The volume of the sphere, V = $\frac{4}{3} \pi R^3$ = $\frac{4}{3} \times \frac{22}{7} \times (22.2)^3 =$ $45848.2 \:cm^3$

Now, finding the error in the volume of the sphere,

$\frac{\triangle V}{V} = 3\frac{\triangle R}{R}$

$\implies \triangle V = V \times 3 \times \frac{\triangle R}{R}$

substituting the values of V, ΔR & R, we get

$\implies \triangle V = 45848.2 \times 3 \times \frac{0.1}{22.2}$

$\implies \triangle V = 619.5\:cm^3$

∴ The volume occupied by the sphere (including the error) is (45848.2 ± 619.5) cm³.

Finding the volume (including error) occupied by the cylinder:

We have,

r = 12.0 cm

Δr = 1.2 cm

h = 24.4 cm

Δh = 1.1 cm

∴ The volume of the cylinder, V = $\pi r^2 h$ = $\frac{22}{7} \times (12)^2\times 24.4 = 11042.7\:cm^3$

Now, finding the error in the volume of the cylinder,

$\frac{\triangle V}{V} = 2\frac{\triangle r}{r} + \frac{\triangle h}{h}$

$\implies \triangle V= V [2\frac{\triangle r}{r} + \frac{\triangle h}{h}]$

substituting the values of V, Δr, r, h & Δh, we get

$\implies \triangle V= 11042.7 [2\frac{1.2}{12} + \frac{1.1}{24.4}]$

$\implies \triangle V= 11042.7 [0.2 + 0.045]$

$\implies \triangle V= 11042.7 \times 0.245$

$\implies \triangle V = 2705.4\:cm^3$

∴ The volume occupied by the sphere (including the error) is (11042.7 ± 2705.4) cm³.

Now,

Finding the total volume occupied by the sphere & cylinder:

∴ The total volume (including the error) is,

= [(45848.2 ± 619.5) cm³] + [(11042.7 ± 2705.4) cm³]

= [45848.2 + 11042.7] ± [619.5 + 2705.4]

= [56890.9 ± 3324.9] cm³

Thus, the total volume occupied by the sphere and by the cylinder (including the error) is [56890.9 ± 3324.9] cm³.

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