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#### karush

##### Well-known member

- Jan 31, 2012

- 3,241

Use differentials to estimate the amount of paint needed

to apply a coat of paint \(\displaystyle 0.05 \text{ cm}\) thick

to a hemispherical dome with a diameter of \(\displaystyle 50\text{ m}\)

$\displaystyle V_h=\frac{1}{2} \cdot\frac{4}{3}\pi\text{r}^3 = \frac{4}{6}\pi\text{r}^3$

$dV_h = 2\pi\cdot r^2\cdot \text{dr}$

so if $$r=25\text{ m} = 2500\text{ cm}\text { and }dr = 0.05\text{ cm}$$ then

$dV_h = 2\pi \ 2500\text{ cm}^2\cdot 0.05\text{ cm}\approx 1.96\text{ m}^3$

or should $r=2500.05\text{ cm}$

to apply a coat of paint \(\displaystyle 0.05 \text{ cm}\) thick

to a hemispherical dome with a diameter of \(\displaystyle 50\text{ m}\)

$\displaystyle V_h=\frac{1}{2} \cdot\frac{4}{3}\pi\text{r}^3 = \frac{4}{6}\pi\text{r}^3$

$dV_h = 2\pi\cdot r^2\cdot \text{dr}$

so if $$r=25\text{ m} = 2500\text{ cm}\text { and }dr = 0.05\text{ cm}$$ then

$dV_h = 2\pi \ 2500\text{ cm}^2\cdot 0.05\text{ cm}\approx 1.96\text{ m}^3$

or should $r=2500.05\text{ cm}$

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